ECE539 - Advanced Theory of Semiconductors and Semiconductor Devices
Numerical Methods and Simulation / Umberto Ravaioli
Introduction to the Numerical Solution of
Partial Differential Equations
1
General concepts
The mathematical models of most physical processes are based on partial differential equations
(PDE). Frequently, a direct analytic solution is possible only for simple cases or under very restrictive assumptions. For the solution of detailed and realistic models, numerical methods are often the
only alternative available. The goal of this chapter is to introduce a number of classical examples,
to familiarize the reader with numerical and practical issues in the solution of PDE’s.
The main objective of a numerical method is to solve a PDE on a discrete set of points of
the solution domain, called discretization. In order to do so, the solution domain is divided into
subdomains having the discretization points as vertices. The distance between two adjacent vertices
is the mesh size. Time is also subdivided into discrete intervals, and we call timestep the interval
between two consecutive times at which the solution is obtained. The PDE is then approximated, or
discretized, to obtain a system of algebraic equations, where the unknowns are the solution values
at the discretization points. This system of algebraic equations can then be solved on a computer
by direct or iterative techniques. It is important to realize that the discretization step replaces
the original equation with a new one, and that even an exact solution of the discretized problem will
yield an approximate solution of the original PDE, since we introduced a discretization error.
A wide variety of equations are encountered in the study of physical phenomena. Equations
have been classified according to the type of phenomena involved, or according to mathematical
features. It is important to attempt a classification, because the choice of a solution approach
depends on the structure of the equation. Typical examples of time-dependent physical processes,
given here in 1-D form, are for instance
Diffusion
Advection or drift
Wave propagation
∂u/∂t = a∂ 2 u/∂x2
∂u/∂t = a∂u/∂x
∂ 2 u/∂t2 = a2 ∂ 2 u/∂x2
For the solution of time-dependent problems of this type (initial value problems), boundary
conditions on the solution domain, as well as the initial condition u(t = 0) are needed. Steady-state
phenomena, characterized by zero time derivatives, can be studied by following the time-dependent
process from an initial condition until a steady-state is reached. In alternative, the steady-state
equations may be solved directly as a boundary value problem, which only requires the knowledge
of boundary conditions. By definition, at steady-state the memory of a specific initial condition is
lost. A typical boundary value problem is
Laplace’s equation
∇2 u = 0
which could be viewed also as the steady-state of a diffusion process. Laplace’s equation is a
particular case of
Poisson’s equation
∇2 u = − ρ/.
1
The classical mathematical classification of elementary PDE’s stems from the analysis of the
general second order PDE
α
∂2u
∂u
∂2u
∂2u
∂u
+ β
+ γ 2 + δ
+ + + ηu = φ
2
∂x
∂x∂y
∂y
∂x
∂y
(1)
The nature of the equation is determined by the coefficients, according to the following classification
a)
b)
c)
β 2 − 4αγ < 0
β 2 − 4αγ = 0
β 2 − 4αγ > 0
Elliptic
(example: Poisson’s equation)
Parabolic
(example: Diffusion equation)
Hyperbolic (example: Wave equation)
Many equations of practical importance may be of a mixed type, or not easily identifiable according to one of the above categories. Nevertheless, the distinction between elliptic, parabolic, and
hyperbolic equations provides a very useful guideline for the selection of solution procedures.
There are many approaches which are used for the discretization of the original PDE to obtain
a numerical problem. The most important discretization approaches can be classified as
• Finite Differences
• Finite Elements
• Spectral Methods
After discretization, it is necessary to check if the approximation is appropriate or if the discretized model can produce a solution at all when programmed into a computer code. For a
successful solution, the numerical scheme must be stable, convergent and consistent.
• The scheme is stable if the solution stays bounded during the solution procedure.
• The scheme is convergent if the numerical solution tends to the real solution as the mesh
size and the timestep tend to zero.
• The scheme is consistent if the truncation error tends to zero as the mesh size and the
timestep tend to zero.
If a numerical scheme is consistent, then stability is a necessary and sufficient condition to achieve
convergence. A scheme which is stable but not consistent may converge to a solution of a different
equation (with which it is consistent).
A number of errors are introduced when a PDE is discretized and solved numerically. To
summarize, we have
• Truncation error - the error introduced by the finite approximation of the derivatives.
• Discretization error - the error in the solution due to the replacement of the continuous
equation with a discretized one.
• Round-off error - the computational error introduced by digital algorithms, due to the
finite number of digits used in the numerical representation.
The round-off error is random in nature, and normally increases when the mesh size is decreased.
Conversely, the discretization error decreases with the mesh size, since more mesh points (i.e. more
resolution) are introduced.
2
2
1-D Finite Differences
The finite difference approach is the most popular discretization technique, owing to its simplicity. Finite difference approximations of derivatives are obtained by using truncated Taylor series.
Consider the following Taylor expansions
u(x + ∆x) = u(x) + ∆x
∂u ∆x2 ∂ 2 u ∆x3 ∂ 3 u
+
+
+ O(∆x4 )
∂x
2 ∂x2
6 ∂x3
∂u ∆x2 ∂ 2 u ∆x3 ∂ 3 u
+
−
+ O(∆x4 )
∂x
2 ∂x2
6 ∂x3
The first order derivative is given by the following approximations:
From (2): Forward Difference
u(x − ∆x) = u(x) − ∆x
∂u
u(x + ∆x) − u(x)
=
+ O(∆x)
∂x
∆x
(2)
(3)
(4)
From (3): Backward Difference
u(x) − u(x − ∆x)
∂u
=
+ O(∆x)
∂x
∆x
By substracting (2) − (3) : Centered Difference
u(x + ∆x) − u(x − ∆x)
∂u
=
+ O(∆x2 )
∂x
2∆x
(5)
(6)
An approximation for the second order derivative is obtained by adding (2) + (3):
∂2u
u(x + ∆x) − 2u(x) + u(x − ∆x)
=
+ O(∆x2 )
∂x2
∆x2
(7)
The terms O(∆x) and O(∆x2 ) indicate the remainders which are truncated (truncation error) to
obtain the approximate derivatives. The centered difference approximation given by (6) is more
precise than the forward difference (4) or the backward difference (5) because the truncation error is
of higher order, a consequence of cancellation of terms of the expansions when taking the difference
between (2) and (3). Since the centered difference involves both neighboring points, there is more
balanced information on the local behavior of the function.
If a better approximation is needed, one could either reduce the size of the mesh or add more
information by including higher order neighbors. It is a good exercise to derive an expression for
the second order derivative including five points instead of three. Besides (2) and (3) we also need
the expansions for u(x + 2∆x) and u(x − 2∆x):
u(x + 2∆x) = u(x) + 2∆x
∂u 4∆x2 ∂ 2 u 8∆x3 ∂ 3 u 16∆x4 ∂ 4 u 32∆x5 ∂ 5 u
+
+
+
+
+ O(∆x6 ) (8)
∂x
2 ∂x2
6 ∂x3
24 ∂x4
120 ∂x5
u(x − 2∆x) = u(x) − 2∆x
∂u 4∆x2 ∂ 2 u 8∆x3 ∂ 3 u 16∆x4 ∂ 4 u 32∆x5 ∂ 5 u
+
−
+
−
+ O(∆x6 ) (9)
∂x
2 ∂x2
6 ∂x3
24 ∂x4
120 ∂x5
The final result is an approximation of order O(∆x4 )
∂2u
−u(x + 2∆x) + 16u(x + ∆x) − 30u(x) + 16u(x − ∆x) − u(x − 2∆x)
≈
∂x2
12∆x2
3
(10)
By including more discretization points, it is also possible to improve the approximation for the
first order derivative. The following are two possible approximations of order O(∆x2 )
∂u
−u(x + 2∆x) + 4u(x + ∆x) − 3u(x)
≈
∂x
2∆x
(11)
∂u
−u(x + 2∆x) + 8u(x + ∆x) − 8u(x − ∆x) + u(x − 2∆x)
≈
(12)
∂x
12∆x
Equation (11) is an improvement of the forward difference and a symmetrical expression can be
derived for the backward difference. Equation (12) is an improvement of the centered difference.
In order to simplify the notation, when convenient we will label the discretization points with
appropriate indices. With ∆x2 = h2 , Eq. (7) becomes
∂2u
u(i + 1) − 2u(i) + u(i − 1)
≈
(13)
2
∂x
h2
Until now, we have assumed a regular grid with uniform mesh intervals, for the derivation of
finite difference approximations. In many cases it may be necessary to discretize the equations
on an irregular grid. Consider a non-uniform 1-D discretization, for example. If finite difference
approximations are sought at the grid point i, we can use the following Taylor expansions:
u(i + 1) = u(i) + hi
∂u h2i ∂ 2 u h3i ∂ 3 u
+
+
+ O(h4i )
∂x
2 ∂x2
6 ∂x3
(14)
∂u h2i−1 ∂ 2 u h3i−1 ∂ 3 u
+
−
+ O(h4i−1 )
(15)
∂x
2 ∂x2
6 ∂x3
The forward and backward difference approximation for the first order derivative can be assembled
as before. An expression which involves both u(i − 1) and u(i + 1) is obtained from (14) − (15) as
u(i − 1) = u(i) − hi−1
∂u
u(i + 1) − u(i − 1)
≈
∂x
hi + hi−1
(16)
Notice that the terms corresponding to the second order derivatives in the Taylor expansions do
not cancel exactly, as in the case of the centered difference in (6) due to the nonuniformity of the
mesh intervals, therefore we have a larger truncation error.
It is often convenient to involve the mid-points of the mesh intervals, indicated as i − 12 and
i + 12 , to express first order derivatives, as
∂u
1
1
hi + hi−1
≈ [u(i + ) − u(i − )]/(
)
(17)
∂x
2
2
2
This result can be useful to obtain an approximation for the second order derivative on the nonuniform mesh. We can write
∂2u
∂ ∂u
∂u
∂u
hi + hi−1
( ) ≈ [ |i+ 1 −
|i− 1 ]/(
)
=
2
∂x
∂x ∂x
∂x 2 ∂x 2
2
(18)
We can then express ∂u/∂x|i+ 1 and ∂u/∂x|i− 1 with centered differences, which are of order O(h2 ),
2
2
and obtain
∂2u
ui+1 − ui ui − ui−1 hi + hi−1 −1
−
](
)
≈[
∂x2
hi
hi−1
2
4
(19)
3
3.1
1-D Examples
Finite Differences for Poisson’s equation
In 1-D, Poisson’s equation is a simple ordinary differential equation. Assuming a uniform dielectric
permittivity , we can write
d2 V (x)
ρ(x)
= −
(20)
2
dx
where V (x) is the unknown electrical potential and ρ(x) is the space dependent charge density.
The equation must be solved on a 1-D domain, where the charge density ρ(x) is sampled at the
discretization points. In order to specify correctly the problem, we need two boundary conditions,
since we are dealing with a second order differential equation. Such boundary conditions could be
constant values for the potential at the boundary points x = 0 and x = L, or one value for the
electric potential V (x) and one value for the electric field E(x), at either boundary point and in
any combination. Remember that the electric field is defined as
∂V (x)
(21)
∂x
It is trivial to show that in a 1-D problem one cannot specify the electric field at the two boundaries,
since in doing so the information on the potential drop across the structure is lost. By using the
finite difference approximation (7), we obtain the discretized 1-D Poisson equation
E(x) = −
ρ(i)
V (i − 1) − 2V (i) + V (i + 1)
= −
(22)
2
∆x
Let’s consider first the case of two potential boundary conditions, e.g. V (1) = V1 and V (N ) =
VN . The domain is divided into N − 1 intervals corresponding to N grid points, of whic 2 are
boundary points, where the solution is known, and N − 2 are internal points, where the solution
is unknown. In order to solve the problem, we need an additional N − 2 equations, besides the
boundary conditions. Therefore, Eq. (22) must be written for each internal boundary point. The
discretization of the differential equation (20) yields a system of N − 2 equations in N unknowns,
reduced to N − 2 unknowns by eliminating the boundary conditions. In order to perform computer
solutions, the system of equations must be represented in matrix form as
AV̄ = B̄
(23)
For the purpose of illustration, let’s consider a uniform mesh with N = 5. The discretized
equations, including boundary conditions are
V (1) = V1
∆x2
ρ(2)
∆x2
V (2) − 2V (3) + V (4) = −
ρ(3)
∆x2
V (3) − 2V (4) + V (5) = −
ρ(4)
V (5) = VN
V (1) − 2V (2) + V (3) = −
This system of equations corresponds to the matrix equation
5







1
0
0
0
1 −2
1
0
0
1 −2
1
0
0
1 −2
0
0
0
0
0
0
0
1
1







V (1)
V (2)
V (3)
V (4)
V (5)


 
 
 
=
 
 
V1
−(∆x2 /)ρ(2)
−(∆x2 /)ρ(3)
−(∆x2 /)ρ(4)
VN







(24)
Since the boundary conditions correspond to two trivial equations, the variables V (1) and V (5)
can be eliminated. The result is the following matrix equation for the internal points





−(∆x2 /)ρ(2) − V1
V (2)
−2
1
0

 


1   V (3)  =  −(∆x2 /)ρ(3)

 1 −2
2
−(∆x /)ρ(4) − VN
V (4)
0
1 −2
(25)
The discretization of Poisson’s equation yields a tridiagonal matrix, where only three diagonals
are non-zero. Matrix equations of this type can be solved quite efficiently, using a fairly compact
variant of Gaussian elimination (Appendix 1).
In many cases, the knowledge of the electric field is also required. By finite differencing the
results for the potential, one can calculate the values of the field on the mesh points using forward,
backward, and centered differences, respectively as
V (i + 1) − V (i)
∆x
V (i) − V (i − 1)
E(i) = −
∆x
V (i − 1) − V (i + 1)
E(i) = −
2∆x
E(i) = −
(26)
(27)
(28)
We know by now that (28) is of order O(∆x2 ) accurate, as opposed to O(∆x) for (26) and (27).
To compute the electric field on a boundary, only (26) and (27) can be used, since the value of the
potential beyond the boundaries is necessary to evaluate the centered difference. In case the value
of the electric field is needed for the mid-points of the mesh intervals, one can use the second order
accurate expressions
1
V (i + 1) − V (i)
E(i + ) = −
(29)
2
∆x
1
V (i) − V (i − 1)
E(i − ) = −
(30)
2
∆x
The 1-D problem can also be solved directly in terms of the electric field. Poisson’s equation is
substituted by Gauss’ law
∂E
ρ(x)
=
(31)
∂x
This is a first order differential equation, and only one boundary condition can be imposed.
The information on the applied potential must be supplied by a condition obtained by integrating
(21) over the domain, which yields
Z L
E(x)dx = V (1) − V (N )
0
In discretized form, (31) becomes
6
(32)
ρ(i − 1)∆x
(33)
where ρ(i − 1) now represents the average charge density inside the mesh interval i − 1. We are
constructing a histogram of the charge contained inside the mesh intervals, and from (33) we obtain
the variation of the electric field from mesh to mesh. If we define
E(i) − E(i − 1) =
∆E(x) = E(x) − E(1)
(34)
we can rewrite (32) as
Z L
0
E(x)dx =
Z L
∆E(x)dx + E(1)L = V (1) − V (N )
(35)
0
On the mesh points we obtain the discretized equation
ρ(i − 1)∆x
(36)
with the boundary condition ∆E(1) = 0. No matrix inversion is needed, because (36) is an explicit
recursion. Once the distribution of ∆E(x) is known, E(1) can be obtained from (35) by performing
a numerical integration. Using Simpson’s formula for the integration (there are of course many
other suitable approaches) we have on a uniform mesh
∆E(i) − ∆E(i − 1) =
"
X
X
V (1) − V (N ) ∆x
E(1) =
∆E(i)
−
∆E(1) + ∆E(N ) + 2
∆E(i) + 4
L
3L
even i
odd i
#
(37)
where the summations are extended to internal mesh points. The distribution of the electric field
on the mesh points is then given by
E(i) = ∆E(i) + E(1)
(38)
Note that the term [V (1) − V (N )]/L in (37) corresponds to the field solution when the charge is
ρ = 0 throughout the domain. An extension of this method to nonuniform mesh simply requires
the use of a different integration procedure in (37)
Going back to Poisson equation, we mentioned above that one boundary condition for the
potential and one boundary condition for the electric field can be specified at either end of the
domain. Let’s assume that we know the conditions E(1) = 0 and V (N ) = VN . We need to choose
an appropriate way to discretize (21) at x = 0, for instance
E(x) = −
∂V (x)
V (2) − V (1)
≈
∂x
∆x
(39)
∂V (x)
V (2) − V (0)
≈
(40)
∂x
2∆x
From (39), the boundary condition E(1) = 0 simply yields V (1) = V (2). In the example with
N = 5, we obtain now the matrix equation
E(x) = −







1 −1
0
0
1 −2
1
0
0
1 −2
1
0
0
1 −2
0
0
0
0
0
0
0
1
1







V (1)
V (2)
V (3)
V (4)
V (5)
7


 
 
 
=
 
 
0
−(∆x2 /)ρ(2)
−(∆x2 /)ρ(3)
−(∆x2 /)ρ(4)
VN







(41)
If the centered difference (40) is choosen, the external point x = − ∆x, with label i = 0 is
also involved in the boundary condition, and we have V (0) = V (2). The potential V (1) at x = 0
is now solved for and we need to add an equation for the boundary point









3.2
1
0 −1
0
0
1 −2
1
0
0
0
1 −2
1
0
0
0
1 −2
1
0
0
0
1 −2
0
0
0
0
0
0
0
0
0
1
1









V (0)
V (1)
V (2)
V (3)
V (4)
V (5)


 
 
 
 
=
 
 
 
0
−(∆x2 /)ρ(1)
−(∆x2 /)ρ(2)
−(∆x2 /)ρ(3)
−(∆x2 /)ρ(4)
VN









(42)
Finite Differences for 1-D Parabolic Equations
We consider here the 1-D diffusion equation
∂u
∂2u
=a
(43)
∂t
∂x2
which is discretized in space and time with uniform mesh intervals ∆x and timestep ∆t. A simple
approach is to discretize the time derivative with a forward difference as
∂u
u(i; n + 1) − u(i; n)
≈
(44)
∂t
∆t
The solution is known at time tn and a new solution must be found at time tn+1 = tn + ∆t.
Starting from the initial condition at t1 = 0, the time evoultion is constructed after each timestep
either explicitly, by direct evaluation of an expression obtained from the discretized equation, or
implicitly, when solution of a system of equations is necessary.
An explicit approach is readily obtained by substituting the space derivative with the 3-point
finite difference evaluated at the current timestep. The algorithm, written for a generic point xi of
the discretization, is
a∆t
[ u(i − 1; n) − 2u(i; n) + u(i + 1; n) ]
(45)
∆x2
A fairly general implicit scheme is obtained by discretizing the space derivative with a weighted
average of the finite difference approximation at tn and tn+1
u(i; n + 1) = u(i; n) +
a∆t
[ u(i − 1; n + 1) − 2u(i; n + 1) + u(i + 1; n + 1) ] =
∆x2
a∆t
= u(i; n) + (1 − λ)
[ u(i − 1; n) − 2u(i; n) + u(i + 1; n) ]
∆x2
u(i; n + 1) − λ
(46)
When λ = 1, the scheme is fully implicit. The classic Crank-Nicholson scheme is obtained when
λ = 0.5 and when λ = 0 the explicit scheme in (45) is recovered. In order to cast the algorithm
in matrix form, all the terms evaluated at time tn+1 are positioned on the left hand side of (46),
while the terms evaluated at time tn are on the right hand side.
Let’s consider again a simple example with N = 5 mesh points and assume that the boundary
conditions u(1; n) = u(1; n+1) = u1 and u(5; n) = u(5; n+1) = u5 are known. The matrix equation
for the discretized diffusion equation is then
8







1
0
0
0
−λr 1 + 2λr
−λr
0
0
−λr
1 + 2λr
−λr
0
0
−λr
1 + 2λr
0
0
0
0
0
0
0
− λr
1







u(1; n + 1)
u(2; n + 1)
u(3; n + 1)
u(4; n + 1)
u(5; n + 1)


 
 
 
=
 
 
u1
RHS(2; n)
RHS(3; n)
RHS(4; n)
u5







(47)
where RHS(i; n) represents the right hand side of (46) and r = a∆t /∆x2 .
Other schemes originate from an evaluation of the time derivative by a centered difference
approximation, in the attempt to reduce the truncation error
2a∆t
[ u(i − 1; n) − 2u(i; n) + u(i + 1; n) ]
(48)
∆x2
This scheme involves three time levels at each step of the iteration, but is explicit and is potentially
much more accurate than the schemes above because of the more precise evaluation of the time
derivative. Unfortunately, better truncation error is not a guarantee for the overall stability of the
scheme. A variant of (48) substitutes u(i; n) with its time average [ u(i; n − 1) + u(i; n + 1) ]/2:
u(i; n + 1) = u(i; n − 1) +
u(i; n + 1) = u(i; n − 1) +
2a∆t
[u(i − 1; n) − u(i; n − 1) − u(i; n + 1) + u(i + 1; n)]
∆x2
(49)
In the next section, the limits of applicability of these discretization schemes are analysed, in
order to determine their validity and usefulness. It should be noted that three-level schemes need
the knowledge of the solution at two consecutive times tn−1 and tn to obtain the solution at tn+1 .
More memory is required as well a special starting procedure.
3.3
Stability analysis
A numerical scheme is unstable when, during the solution procedure, the result becomes unbounded
eventually reaching overflow, independently of the choice of the computing system. The phenomenon of instability arises when the numerical procedure itself tends to amplify errors (e.g.
round-off error, which is unavoidable on any digital computer) to the point that the amplified error
obliterates the solution itself. This may happen even after derivatives are replaced by approximations which are more accurate from a numerical point of view.
We examine here an initial value problem of the type
∂u
= Lu
(50)
∂t
where L is a differential space operator. The equation is solved at discrete times separated by a
timestep ∆t. Discretization of (50) will yield a matrix equation of the type
A ū(t = (n + 1)∆t) = B ū(t = n ∆t)
(51)
The solution at a given time is related to the solution at a previous timestep through a transformation of the type
ū(t = (n + 1)∆t) = T ū(t = n ∆t)
(52)
where ū is the vector of solutions obtained on the discretization points, and T = A−1 B is a
transformation matrix which governs the time evolutions, according to the discretized numerical
scheme. Starting from the initial condition, the iteration evolves in the following way
9
ū(∆t) = T ū(0)
ū(2∆t) = T ū(∆t) = T2 ū(0)
ū(3∆t) = T ū(2∆t) = T3 ū(0)
(53)
......
n
ū((n + 1)∆t) = T ū(n ∆t) = T ū(0)
The eigenvalues λi and the eigenvectors v̄i of the matrix T are related through
T v̄i = λi v̄i
(54)
Note that there are as many eigenvalues and eigenvectors as internal mesh points in the discretized
domain. The solution at a given timestep can be expressed as a linear combination of the eigenvectors. At t = 0 we have
ū(0) =
X
cn v̄i
(55)
i
with i = 1, N being N the total number of internal mesh points. The last equation in (53) can be
rewritten as
ū[(n + 1)∆t] = Tn ū(0) =
X
ci (λi )n v̄i
(56)
i
The solution stays bounded if the modulus of the largest eigenvalue | λi |max (spectral radius)
of the matrix T satisfies the condition | λi |max ≤ 1.
In many cases of practical interest, stability analysis can be performed with a Fourier method.
Fourier analysis is performed by expanding the solution in space Fourier components over the
discretization. Then we can just study the evolution of a generic Fourier component (mode)
between timesteps, since modes are orthogonal and independent of each other. A Fourier mode is
expressed as
Fω (i; n) = W (n) exp ( Ω xi )
(57)
where  is the imaginary unit, Ω is the Fourier mode wavenumber, and W (n) is the mode amplitude.
The indices i and n indicate mesh point and timestep, respectively.
Stability is analyzed by replacing the variable u with the generic Fourier component in the
discretized scheme. The discretization is stable if the mode amplitude remains bounded at any
iteration step. This is verified if
W (n + 1)
≤ 1 + O(∆t)
W (n)
(58)
The Fourier analysis does not include the effect of boundaries, which may affect the actual stability
behavior. Boundary conditions can be introduced as additional equations when the eigenvalues of
the time-evolution matrix are analyzed.
10
3.4
Stability analysis for the diffusion equation
(a) Explicit schemes
Consider (45) and substitute u(i; n) with the Fourier mode Fω (i; n)
W (n + 1) exp( Ωxi ) = W (n) exp( Ωxi ) +
a∆t
W (n) [ exp( Ωxi+1 ) − 2 exp( Ωxi ) + exp( Ωxi−1 ) ]
∆2 x
(59)
Since exp( Ωxi±1 ) = exp( Ωxi ) exp(±Ω∆x), (59) becomes
W (n + 1)
a∆t
= 1 +
[ exp( Ω∆x) − 2 + exp(−Ω∆x) ]
W (n)
∆2 x
(60)
W (n + 1)
4a∆t
= 1 −
sin2 (Ω∆x/2)
W (n)
∆2 x
(61)
For stability, we must have | W (n + 1)/W (n) | ≤ 1. This inequality is satisfied only when
4a∆t
sin2 (Ω∆x/2) ≤ 2
(62)
∆2 x
A physical diffusion constant is a > 0, therefore the left hand side of (62) is always positive. Since
0 ≤ sin2 (Ω∆x/2) ≤ 1, we can restrict ourselves to the worst case sin2 (Ω∆x/2) = 1. The
stability condition becomes
4a∆t
∆2 x
≤ 2
(63)
which gives the condition for the timestep
∆2 x
(64)
2a
This result indicates that once the mesh size ∆x is fixed, there is a limit in the choice of the timestep
∆t. The timestep can be increased beyond this limit, only if the mesh size is also increased, leading
to a reduced space resolution.
The same approach can be used to analyze the stability of the three-level scheme (48) but
the analysis is slightly more difficult now. For the generic Fourier mode we obtain an expression
analogous to (61)
∆t ≤
W (n + 1)
W (n − 1)
8a∆t
=
−
sin2 (Ω∆x/2)
W (n)
W (n)
∆2 x
(65)
If we define W (n + 1)/W (n) = W (n)/W (n − 1) = g, we obtain the quadratic equation
g2 + g
8a∆t
sin2 (Ω∆x/2) − 1 = 0
∆2 x
(66)
with solution
4a∆t
g = −
sin2 (Ω∆x/2) ±
∆2 x
s
1 +
11
16a2 ∆2 t
sin4 (Ω∆x/2) = 0
∆4 x
(67)
It is easy to see that the solution with negative sign in front of the square root leads to |g| > 1
for any choice of finite ∆t and ∆x. Therefore, the scheme (48) is unconditionally unstable! The
use of a more accurate approximation for the time derivative does not produce a better difference
approximation for the whole equation.
The modified scheme in (49) is instead always stable. However, an analysis of the local truncation error reveals that the scheme has a consistency problem.
(b) Implicit schemes
The Fourier stability method applied to (46) yields the equation
a∆t
W (n + 1)
1 −
λ [ exp( − Ω∆x) + exp( Ω∆x) − 2 ] =
∆2 x
a∆t
(1 − λ) [ exp( − Ω∆x) + exp( Ω∆x) − 2 ]
W (n)
1 +
∆2 x
(68)
(69)
from which we have
W (n + 1)
=
W (n)
1 − 4
a∆t
∆2 x
1 + 4
(1 − λ) sin2 ( Ω∆x/2)
a∆t
∆2 x
λ sin2 ( Ω∆x/2)
(70)
and for stability
1 −
4
1 +
a∆t
sin2 ( Ω∆x/2)
∆2 x
λ sin2 ( Ω∆x/2)
4 a∆t
∆2 x
≤ 1
(71)
When λ = 0 we recover the stability condition found previously for the explicit scheme (45). If we
set sin2 (Ω∆x/2) = 1, we have the stability condition
λ ≥
1
∆2 x
−
2
4 a∆t
!
(72)
From this result, we can see that both the Crank-Nicholson method (λ = 0.5) and the fully
implicit method (λ = 1) are unconditionally stable and any timestep ∆t can be chosen. Of
course, stability is no guarantee that the physical evolution will be properly simulated, and the
timestep should be selected so that it does not exceed characteristic time constants typical of the
process under investigation.
3.5
Explicit Finite Difference Methods for the Advection Equation
The advection equation has the form
∂u(x)
∂vu(x)
+
= 0
(73)
∂t
∂x
where v has the dimensions of a velocity and may be space-dependent. The advection equation is
hyperbolic in nature and describes phenomena of propagation which should be solved in a natural
way by obtaining the trajectories of constant solution in the space-time domain. The locus of
constant solution is called characteristic. Characteristics methods for the solution are in general
more expensive than finite difference methods. In particular, explicit difference methods are extremely simple to code and fast, although one has to make sure that the physics of the problem is
well represented.
12
We can try to discretize (73) by simply using a forward difference for the time and a centered
difference for the space derivatives as
∆t
[ v(i + 1) u(i + 1; n) − v(i − 1) u(i − 1; n) ]
(74)
2∆x
Unfortunately, stability analysis shows that this scheme is unconditionally unstable. Some improvement is obtained by subsituting u(xi , tn ) on the right hand side with a space average
u(i; n + 1) = u(i; n) −
u(i; n + 1) =
u(i − 1; n) + u(i + 1; n)
+
2
∆t
[ v(i + 1) u(i + 1; n) − v(i − 1) u(i − 1; n) ]
2∆x
(75)
The scheme is now conditionally stable. Assuming for simplicity a constant velocity v, the stability
condition is ∆t ≤ ∆x/|v|. This result is usually called C.F.L. condition (from the mathematicians Courant, Friederichs and Lewy). Since ∆x/∆t has the dimensions of a velocity, the choice of
discretization mesh imposes a characteristic velocity ∆x/∆t which cannot be exceeded in order to
obtain physical results.
There are some subtle details which are not evident from a simple stability analysis. Let’s
rewrite the previous schemes, adding u(i; n − 1)/2 − u(i; n − 1)/2 = 0 to the left hand side of (74),
and u(i; n − 1) − u(i; n − 1) + u(i; n) − u(i; n) = 0 to the left hand side of (75). Using a constant
velocity, (74) becomes
u(i; n + 1) − u(i; n − 1)
u(i; n + 1) − 2u(i; n) + u(i; n − 1)
+
=
2
2
v∆t
−
[ u(i + 1; n) − u(i − 1; n) ]
2∆x
(76)
which is a second order discretization of the equation
∆t ∂ 2 u(x)
∂u(x)
∂u(x)
+
+ v
= 0
2
∂t
2
∂t
∂x
(77)
and we obtain for (75)
u(i; n + 1) − u(i; n − 1)
u(i; n + 1) − 2u(i; n) + u(i; n − 1)
+
=
2
2
(78)
u(i + 1; n) − 2u(i; n) + u(i − 1; n)
v∆t
−
[ u(i + 1; n) − u(i − 1; n) ]
2
2∆x
which is in turn a second order discretization of the equation
∂u(x)
∆t ∂ 2 u(x)
∆x2 ∂ 2 u(x)
∂u(x)
+
−
+ v
= 0
(79)
2
2
∂t
2
∂t
2∆t ∂x
∂x
The stabilization in (75) is obtained by the addition of a diffusive term which is able to quench the
growth of instabilities as the solution evolves in time. The introduction of this spurious diffusion
causes smoothing, due to the coupling introduced between space Fourier components (modes) of the
solution, which modifies the shape of the traveling solution itself. In addition, the discretization may
13
cause the modes to travel at different velocity (dispersion), also causing a spurious deformation
of the solution. When |v| = ∆x/∆t, dispersion is eliminated, but obviously this holds only in the
case of uniform velocity.
A stable discretization of (73) is obtained with first order space differences, using a forward
difference when the velocity is positive and a backward difference when the velocity is negative
u(i; n + 1) = u(i; n) −
∆t
[ v(i + 1) u(i + 1; n) − v(i) u(i; n) ] ;
∆x
v>0
(80)
∆t
= u(i; n) +
[ v(i) u(i; n) − v(i − 1) u(i − 1; n) ] ;
∆x
v<0
This procedure is called upwinding, since the direction is adjusted according to the direction of
the flow.
A second order scheme can be obtained by using a centered time difference, which leads to a
three-level time scheme, called leapfrog method
∆t
[ v(i + 1) u(i + 1; n) − v(i − 1) u(i − 1; n) ]
(81)
∆x
This method is stable if the C.F.L. condition is satisfied. Note that, as shown earlier, a similar
method applied to the diffusion equation yields an unstable scheme.
A second order scheme can be obtained considering the Taylor expansion in time
u(i; n + 1) = u(i; n − 1) −
∂u
∆t2 ∂ 2 u
+
+ O(∆t3 )
∂t
2 ∂t2
The time derivatives can be expressed from the advection equation itself as
u(x, t + ∆t) = u(x, t) + ∆t
(82)
∂vu(x)
∂u(x)
∂v(x)
∂u(x)
∂u(x)
= −
= − v(x)
− u(x)
≈ − v(x)
∂t
∂x
∂x
∂x
∂x
(83)
∂ 2 u(x)
∂
∂u(x)
= − v(x)
v(x)
2
∂t
∂x
∂x
(84)
where we have assumed that the velocity is a slow varying space variable and that we can neglect
the term u(x) ∂v(x)/∂x. The second order Lax-Wendroff scheme is then obtained by discretizing
(82)
u(i; n + 1) = u(i; n) −
v(i)∆t
[v(i + 1)u(i + 1; n) − v(i − 1)u(i − 1; n)] −
2∆x
(85)
v(i)∆2 t
2∆2 x
{v(i + 1/2) [u(i + 1; n) − u(i; n)] − v(i − 1/2) [u(i; n) − u(i − 1; n)]}
The last term on the right hand side of (85) corrects the instability of the scheme, and is called
artificial viscosity. Note that the velocity must be known at the midpoint of the mesh intervals.
14
4
2-D Finite Differences
In the case of a 1-D domain, there are only two possible discretization approaches: uniform or
nonuniform mesh intervals. A 2-D domain can be decomposed by using any combination of polygons having the discretization nodes as vertices. Finite difference methods are very convenient in
conjunction with rectangular grids, although finite difference approximations of differential operators can be written for general meshes. Finite difference discretizations for a rectangular grid
are usually assembled by considering the four neighbors of the current mesh point, although other
schemes are possible.
It is relatively easy to develop 2-D discretization on a general rectangular grid, starting from
the results for the 1-D case. First order derivatives evaluated along the cartesian directions x and
y are differenced in the same way, and the variables are now denoted by two indices. A finite
difference approximation for the 2-D laplacian is easily assembled on a uniform (square) mesh,
where ∆x = ∆y = ∆, by combining two expressions like (7) written for the x− and y−direction
∂ 2 u(x, y)
∂ 2 u(x, y)
+
=
∂x2
∂y 2
u(i + 1, j) − 2u(i, j) + u(i − 1, j) u(i, j + 1) − 2u(i, j) + u(i, j − 1)
+
+ O(∆2 ) =
∆2
∆2
u(i + 1, j) + u(i − 1, j) − 4u(i, j) + u(i, j + 1) + u(i, j − 1)
+ O(∆2 )
∆2
∇2 u =
(86)
This finite difference approximation involves the current point and the four nearest neighbors
in the square grid (5-point formula). To improve accuracy, it is possible to involve also the second
nearest neighbors (9-point formula) using the 1-D result in (10)
∂ 2 u(x, y)
∂ 2 u(x, y)
+
= [−ui+2,j − ui,j+2 + 16ui+1,j + 16ui,j+1 − 60ui,j +
∂x2
∂y 2
(87)
2
4
16ui−1,j + 16ui,j−1 − ui−2,j − ui,j−2 ]/(12∆ ) + O(∆ )
On a nonuniform rectangular grid, we can use (19) with ∆x(i) = hi and ∆y(j) = kj to obtain
a 5-point formula which is accurate to first order
∂ 2 u(x, y)
∂ 2 u(x, y)
+
≈
∂x2
∂y 2
ui+1,j − ui,j
ui,j − ui−1,j
−
hi
hi−1
ui,j+1 − ui,j
ui,j − ui,j−1
−
kj
kj−1
#
/(
hi + hi−1
)
2
(88)
"
+
/(
kj + kj−1
)
2
We can obtain the result in (88) with a more general approach called box-integration. Consider the box around the generic mesh point (xi , yj ) obtained after bisecting the mesh lines connecting the point to its neighbors. The differential operator is integrated over the volume of the
box, which in 2-D is just the box area
Abox =
hi−1 + hi kj−1 + kj
2
2
15
(89)
The integral of the Laplacian over the domain D of the box can be converted into an integral over
the boundary ∂D of the domain by using Green’s theorem
Z Z
2
∇ u(x, y) dx dy =
Z Z
D
∇ · ∇u(x, y) dx dy =
D
Z yj + kj
2
=
yj −
−
Z yj − kj−1
2
yj +
kj
2
kj−1
2
Z
∂D
∂
hi
u xi + , y dy +
∂x
2
∂
hi−1
u xi −
, y dy −
∂x
2
Z xi − hi−1
2
xi +
hi
2
Z xi + hi
2
xi −
hi−1
2
∂u
∂u
dy −
dx
∂x
∂y
∂
kj
u x, yj +
∂y
2
dx
(90)
∂
kj−1
dx
u x, yj −
∂y
2
The derivatives at the mesh midpoints, which appear inside the line integrals in (90) can be
approximated by using centered finite differences involving the neighboring mesh points. These
values are used as an approximation of the average integrands for the evaluation of the line integrals,
which are obtained by multiplying the average value by the corresponding length of the integration
path, for instance:
Z yj + kj
2
yj −
kj−1
2
∂
hi
u(i + 1, j) − u(i, j) kj−1 + kj
u xi + , y dy ≈
∂x
2
hi
2
(91)
We then evaluate
Z Z
D
2
hi−1 + hi kj−1 + kj
= ∇ · ∇u(i, j)
2
2
2
∇ u(x, y) dx dy ≈ ∇ u(i, j)Abox
(92)
where we have assumed that the value of the laplacian on the mesh point is a good approximation for
the average value inside the box, and we obtain finally the following finite difference approximation
for the laplacian
u(i + 1, j) − u(i, j) kj−1 + kj
+
hi
2
u(i, j + 1) − u(i, j) hi−1 + hi u(i − 1, j) − u(i, j) kj−1 + kj
+
kj
2
hi−1
2
∇ · ∇u(i, j) ≈
# u(i, j − 1) − u(i, j) hi−1 + hi
hi−1 + hi kj−1 + kj
+
/
kj−1
2
2
2
(93)
It can be easily verified that the finite difference approximations (88) and (93) are identical. The
box integration method provides a rigorous approximation procedure which is very useful for the
discretization of complicated operators. The box integration is not limited to finite differences.
Note that, up to (90), no approximation has been made and any suitable discretization method
can be used from that point on.
4.1
2-D Finite Differences for Poisson’s equation
In the case of constant dielectric permittivity , the 2-D Poisson’s equation is
∂ 2 V (x, y)
ρ(x, y)
∂ 2 V (x, y)
+
=
2
2
∂x
∂y
16
(94)
The 5-point finite difference discretization of the 2-D Poisson’s equation is given directly by (86)
for uniform square mesh
V (i + 1, j) + V (i − 1, j) − 4V (i, j) + V (i, j + 1) + V (i, j − 1)
ρ(i, j)
=−
2
∆
or by (88) for nonuniform rectangular mesh
Vi,j − Vi−1,j
Vi+1,j − Vi,j
−
hi
hi−1
/
hi + hi−1
2
(95)
(96)
"
+
Vi,j+1 − Vi,j
Vi,j − Vi,j−1
−
kj
kj−1
# /
kj + kj−1
2
=−
ρ(i, j)
The indexing in (95) and (96) assumes a natural mesh point numeration according to columns (i)
and rows (j) of a rectangular grid. The scheme is often called lexicographic because the indexing
follows the usual pattern of writing. An equation like (95) or (96) is written for each internal node
in the 2-D mesh. On the boundary points we either specify a fixed potential (Dirichlet boundary
condition) or the normal component of the electric field (von Neuman boundary condition). A
Dirichlet boundary condition must be imposed at least at one boundary point for the problem to
be specified.
The ordering of the mesh points affects the structure of the discretized matrix. For illustration,
let’s consider a simple rectangular domain discretized with a uniform square mesh, so that ∆x =
∆y = ∆, and i = 1, 2, 3, 4, 5 and j = 1, 2, 3, 4. For simplicity, fixed potential boundary conditions
are imposed on all boundary points. Following the lexicographic ordering, we have equations like:
Row 1:
V (2, 1) = V21
V (3, 1) = V31
V (4, 1) = V41
Row 2:
V (1, 2) = V12
∆2 ρ(2, 2)
V (2, 1) + V (1, 2) − 4V (2, 2) + V (3, 2) + V (2, 3) = −
∆2 ρ(3, 2)
V (3, 1) + V (2, 2) − 4V (3, 2) + V (4, 2) + V (3, 3) = −
∆2 ρ(4, 2)
V (4, 1) + V (3, 2) − 4V (4, 2) + V (5, 2) + V (4, 3) = −
V (5, 2) = V52
Row 3:
17
V (1, 3) = V13
∆2 ρ(2, 3)
V (2, 2) + V (1, 3) − 4V (2, 3) + V (3, 3) + V (2, 4) = −
∆2 ρ(3, 3)
V (3, 2) + V (2, 3) − 4V (3, 3) + V (4, 3) + V (3, 4) = −
∆2 ρ(4, 3)
V (4, 2) + V (3, 3) − 4V (4, 3) + V (5, 3) + V (4, 4) = −
V (5, 3) = V53
Row 4:
V (2, 4) = V24
V (3, 4) = V34
V (4, 4) = V44
Notice that the boundary conditions for the corner points V (1, 1), V (5, 1), V (1, 4) and V (5, 4)
have not been included, because they do not enter the expressions for the internal points. After
elimination of the boundary conditions, the resulting matrix equations is










−4
1
0
1
0
0
1 −4
1
0
1
0
0
1 −4
0
0
1
1
0
0 −4
1
0
0
1
0
1 −4
1
0
0
1
0
1 −4









V (2, 2)
V (3, 2)
V (4, 2)
V (2, 3)
V (3, 3)
V (4, 3)


 
 
 
 
=
 
 
 


2
− ∆ ρ(2,2)
2
− ∆ ρ(3,2)
2
− ∆ ρ(4,2)
∆2 ρ(2,3)
− 2
− ∆ ρ(3,3)
∆2 ρ(4,3)
− − V21 − V12
− V31
− V41 − V52
− V13 − V24
− V34
− V44 − V53












(97)
The discretization matrix presents five non-zero diagonals. The system of linear equations can be
solved by a direct technique (e.g. Gaussian elimination) or by using an iterative procedure.
4.2
Iterative schemes for the 2-D Poisson’s equation
The general idea of iterative schemes is to start with an initial guess for V (i, j) and then apply
repeatedly a correction scheme, which modifies the solution at each iteration, until convergence is
obtained. The goal of an iterative scheme is to arrive at a solution which approximates, within a
given tolerance, the exact solution of the discretized equations. Direct methods, instead, attempt
an exact solution of the discretized problem, with accuracy limited only by the round-off error
An iterative procedure is similar to a time-dependent problem which evolves to steady-state
from an initial solution, and therefore the iteration scheme must stable and convergent. Since the
actual solution cannot be exactly obtained in practice, there is the need of a stopping criterion to
decide when to interrupt the iterations. At each iteration we define the error as
(i, j; k) = | u∗ (i, j) − u(i, j; k) |
(98)
where u∗ (i, j) is the true solution of the discretized problem at a given mesh point (i, j) and u(i, j; k)
is the solution at iteration k. A scheme is stable if the error tends to zero asymptotically. Remember
that for our purposes zero is the tolerance fixed in the beginning. The error cannot in gneral reach
18
exactly zero because of noise introduced by round-off error. In some schemes, it is also possible for
the error to increase temporarily before starting to decrease. In order to apply (98) directly, one
needs to know already the solution, which is usually not the case except when the code output is
compared to some known results for testing purposes. A simple way to test the convergence is to
calculate for every discretized equation the residuals, obtained by replacing the solution into the
discretization. For uniform discretization, we have from (95)
R(i, j; k) = [u(i + 1, j; k) + u(i − 1, j; k) − 4u(i, j; k) + u(i, j + 1; k)
(99)
+u(i, j − 1; k)]/∆2 +
ρ(i, j)
Obviously, the residuals tend to zero as the error (98) tends to zero, and the iterations can be
stopped when the residuals are sufficiently low.
In order to formulate an effective stopping criterion, the norm of local errors or residuals is
used. There are various possible formulations of the norm, as defined in Hilbert spaces. We will
consider here the following two norms
1/2

X
kk2 =  2 (i, j; k)
(100)
ij
kk∞ = max{(i, j; k)}
(101)
Similar norms can be written for the residuals. The expression in (100) simply corresponds to the
root mean square of the local error, and in the language of Hilbert spaces is also called `2 norm.
The expression in (101) is the `∞ norm and is obtained by searching for the maximum value. The
iterative process is stopped when the selected norm is smaller than the tolerance. The `2 norm
gives a more robust stopping criterion.
If it is known that the iterative scheme is convergent, one can examine the difference between
the solutions at two different steps. Convergence is assumed when the norm
1/2

X
k∆uk2 =  (u(i, j; k + 1) − u(i, j; k))2 
(102)
ij
or
k∆uk∞ = max{u(i, j; k + 1) − u(i, j; k)}
(103)
is smaller than the selected tolerance. This is possible because, as convergence is approached, the
local variations of the iterative solution become smaller and smaller.
In the following, we report a number of classical schemes for the iterative solution of the 2-D
Poisson equation discretized with finite differences using a uniform mesh ∆
4.2.1
Jacobi iteration
u(i, j; k + 1) = u(i, j; k) + [u(i, j − 1; k) + u(i − 1, j; k) − 4u(i, j; k)
(104)
2
+u(i + 1, j; k) + u(i, j + 1; k) + ρ(i, j)∆ /]/4
19
The scheme is written in a slightly redundant way, to show that the updated value is equal to the
previous one plus a correction which, for this particular scheme, corresponds to one fourth of the
residual calculated at the previous iteration. We already know that the residual must tend to zero
as the solution is approached.
The scheme in (105) can be rewritten in the compact form
u(i, j; k + 1) = [u(i, j − 1; k) + u(i − 1, j; k) + u(i + 1, j; k) + u(i, j + 1; k)
(105)
2
+ρ(i, j)∆ /]/4
In matrix form, we can formally write
∆2
ρ̄
(106)
4
where B is the Jacobi matrix. Note that the Jacobi iteration requires at each time step enough
memory space for the solution at both iteration k and k + 1.
ū(k + 1) = Bū(k) +
4.2.2
Gauss–Seidel iteration
u(i, j; k + 1) = u(i, j; k) + [u(i, j − 1; k + 1) + u(i − 1, j; k + 1) − 4u(i, j; k)
(107)
2
+u(i + 1, j; k) + u(i, j + 1; k) + ρ(i, j)∆ /]/4
This scheme is identical to the Jacobi iteration, except that solutions at points which are scanned
before (i, j) are immediately updated. From a computational point of view, this is automatically
accomplished if one single array is used to store both old and updated solution. The Gauss–Seidel
iteration has an asymptotic convergence rate which is slightly better than the Jacobi iteration.
4.2.3
Successive Under Relaxations (SUR)
u(i, j; k + 1) = u(i, j; k) +
ω
[u(i, j − 1; k) + u(i − 1, j; k) − 4u(i, j; k)
4
(108)
2
+u(i + 1, j; k) + u(i, j + 1; k) + ρ(i, j)∆ /]
The SUR scheme is a modification of the Jacobi iteration, where the correction term is multiplied
by an underrelaxation parameter ω. For convergence, it must be 0 < ω ≤ 1. When ω = 1 the
Jacobi iteration is recovered.
4.2.4
Successive Over Relaxations (SOR)
u(i, j; k + 1) = u(i, j; k) +
ω
[u(i, j − 1; k + 1) + u(i − 1, j; k + 1) − 4u(i, j; k)
4
(109)
2
+u(i + 1, j; k) + u(i, j + 1; k) + ρ(i, j)∆ /]
20
The SOR scheme is a modification of the Gauss–Seidel iteration, where the correction term is
multiplied by an overrelaxation parameter ω. For convergence, it must be 1 ≤ ω < 2. The Gauss
Seidel scheme is recovered when ω = 1. The best convergence rate is obtained with an optimum
relaxation parameter
ωopt =
2
1 + (1 − µ2 )1/2
(110)
where λ is the spectral radius (modulus of largest eigenvalue) of the Jacobi matrix B.
The SOR iteration has considerably better convergence properties than the Gauss–Seidel iteration. However, in the first few iterations the error actually grows, to fall sharply afterwards, while
the Gauss–Seidel iteration has considerable error reduction during the first few iterations.
4.2.5
Cyclic (Red-Black) SOR
The grid points are subdivided into two partitions, as the red and black squares on a checkerboard.
The Cyclic SOR iteration operates in two steps:
• Update the values on the red grid points using the SOR step
ured (i, j; k + 1) = ured (i, j; k) +
ω
[ubl (i, j − 1; k) + ubl (i − 1, j; k)
4
(111)
2
−4ured (i, j; k) + ubl (i + 1, j; k) + ubl (i, j + 1; k) + ρ(i, j)∆ /]
• Update the values on the black grid points, using the SOR step
ubl (i, j; k + 1) = ubl (i, j; k) +
ω
[ured (i, j − 1; k + 1) + ured (i − 1, j; k + 1)
4
(112)
2
−4ubl (i, j; k) + ured (i + 1, j; k + 1) + ured (i, j + 1; k + 1) + ρ(i, j)∆ /]
The optimum overrelaxation parameter is the same as ωopt found for the SOR iteration. Note that
at each of the two steps the partitions of red and black points are independent, since the solution
is only updated on points with the same color.
4.2.6
Cyclic Chebishev iteration
The Cyclic Chebishev iteration is a variation of the Cyclic SOR, where the overrelaxation parameter
is varied during the iterations, according to the following scheme
• Iterate on red points with ω(1) = 1
• Iterate on black points with ω(2) = (1 − µ2 /2)−1 . . .
• Iterate on red points with ω(k) = (1 − ω(k − 1)µ2 /4)−1
• Iterate on black points with ω(k + 1) = (1 − ω(k)µ2 /4)−1 . . .
In the limit, ω∞ → ωopt . At the first step, the scheme behaves like the Gauss–Seidel iteration, with
good error reduction.
21
4.2.7
Line Iterations
Line iterations are obtained by processing at one time an entire line (row) of N grid points, and
the resulting implicit problem requires the solution of a system of N equations. Line iterations are
possible for all the point iterations listed above. The Jacobi line iteration has the form
1
u(i, j; k + 1) = [u(i + 1, j; k + 1) + u(i − 1, j; k + 1)
4
(113)
2
+u(i, j − 1; k + 1) + u(i, j + 1; k) − ρ(i, j)∆ /]
The scheme in (114) is similar to the point Jacobi iteration as given by (106), but here the variables
corresponding to grid points on the current row are unknowns and are to be solved simultaneously
at iteration (k + 1). By placing the unknowns on the left hand side, we obtain
−u(i − 1, j; k + 1) + 4u(i, j; k + 1) − u(i + 1, j; k + 1) =
(114)
2
u(i, j − 1; k) + u(i, j + 1; k) − ρ(i, j)∆ /]
The values at nodes on adjacent rows are evaluated at the previous iteration k. The set of equations
obtained for the grid points of the row has a tridiagonal matrix, and can be solved with the method
described earlier. The Gauss-Seidel line iteration is obtained simply by replacing u(i, j − 1; k) with
u(i, j − 1; k + 1), since the latter value is already available from the iteration for row (j − 1)
−u(i − 1, j; k + 1) + 4u(i, j; k + 1) − u(i + 1, j; k + 1) =
(115)
2
u(i, j − 1; k + 1) + u(i, j + 1; k) − ρ(i, j)∆ /]
The Successive Line Over Relaxation (SLOR) method is obtained by applying a relaxation to
the Gauss Seidel line iteration
−u(i − 1, j; k + 1) + 4u(i, j; k + 1) − u(i + 1, j; k + 1) =
+(1 − ω)[4u(i, j; k) − u(i + 1, j; k) − u(i − 1, j; k)]
(116)
2
+ω[u(i, j − 1; k) + u(i, j − 1; k + 1) − ρ(i, j)∆ /]
Line iteration schemes usually converge faster and require less iterations than the corresponding
point iteration schemes. The coding is somewhat more difficult, and while the point iteration
requires N solution evaluations per row, the line iteration requires the solution of an N × N
tridiagonal matrix. The iteration cost may be comparable, provided the line iteration is coded
carefully. The idea behind line iterations can be extended to block iteration, where a group
(block) of adjacent rows are solved simultaneously.
4.2.8
Alternating-Direction Implicit (ADI) Iteration
This iteration belongs to the category of operator splitting methods. Rather than solving the
full implicit method at once, the horizontal and vertical components of the laplacian operator
22
are solved implicitly in two consecutive 1-D steps along the rows and along the columns of the
grid. Since two 1-D solutions are in general not equivalent to the full 2-D solution, the procedure
is iterated and convergence is aided by the application of a relaxation parameter β.
a) Horizontal sweep
−u(i − 1, j; k + 1/2) + (2 + β)u(i, j; k + 1/2) − u(i + 1, j; k + 1/2) =
(117)
2
u(i, j − 1; k) − (2 − β)u(i, j; k) + u(i, j + 1; k) + ρ(i, j)∆ /
Every row is solved implicitly generating a tridiagonal system of equations, Similarly to the line
iteration schemes. Since the values at the previous iteration are used to compute the vertical derivatives, the tridiagonal systems are all independent of each other and can be solved simultaneously.
This step of the iteration resembles the Jacobi line iteration, however, here 1-D differential operators are considered for the implicit (horizontal) and the explicit (vertical) terms, and a relaxation
parameter β is also used.
b) Vertical sweep
−u(i, j − 1; k + 1) + (2 + β)u(i, j; k) − u(i, j + 1; k + 1) =
(118)
2
u(i − 1, j; k + 1/2) − (2 − β)u(i, j; k + 1/2) + u(i + 1, j; k + 1/2) + ρ(i, j)∆ /
Now, the same step is applied to columns of the grid, and the values at the previous k + 1/2 step
are used to express horizontal derivatives. Note that results obtained after the first step are very
biased, and a convergence check should only be applied after both steps have been performed. If
the relaxation parameter β is taken to be equal to ωopt , ADI has the same convergence rate of SOR.
4.2.9
Cyclic ADI Iteration
To speed up the convergence rate of ADI, the relaxation parameter β is varied cyclically at every
iteration k. For a rectangular domain, with N and M mesh intervals along the x− and y−sides,
the values of β(k) are found according to the following algorithm
a)
b)
c)
d)
Select
Define
s = max{M + 1, N + 1}
a = 4 sin2 (π/2s)
b = 4 sin2 [(s − 1)π/2s]
c = a/b √
δ =3−2 2
Find the smallest integer m, such that δ m−1 ≤ c
Calculate the cyclic relaxation parameters
β(i) = bcν(i) with ν(i) = (i − 1)/(m − 1); 1 ≤ i ≤ m
When more than m iterations are necessary, the sequence of parameters β(i) is repeated. With the
application of cyclical relaxation parameters, convergence is normally achieved in a much smaller
number of iterations than with an equivalent SOR scheme.
23
4.2.10
Matrix Notation for the Iterative Algorithms
Linear algebra notation is often used to represent in a compact form the iterative algorithms
discussed above. The discretized Poisson equation can be represented by the linear system
Aū = ḡ
(119)
where the discretization matrix A is obtained as in (97). The matrix can be decomposed as
A=L+U+D
(120)
where D is the diagonal, and L and U are respectively the lower triangular and the upper
triangular parts. The Jacobi iteration correspond to the matrix equation
Dū(k + 1) = −(L + U)ū(k) + ḡ
(121)
ū(k + 1) = −D−1 [L + U]ū(k) + D−1 ḡ
(122)
or
where −D−1 (L + U) is the Jacobi matrix.
In terms of the elements ai,j of the original matrix A, the Jacobi matrix for a rectangular
domain with Dirichlet boundary conditions has the form




−D−1 (L + U) = − 
0
a12 /a11 a13 /a11
a21 /a22
0
a23 /a22
...
...
...
an1 /ann an2 /ann an3 /ann
. . . a1n /a11
. . . a2n /a22
...
...
...
0





(123)
where n = N × M , with N and M the internal grid nodes on the sides of the domain. It can be
shown that for this simple case the spectral radius (i.e. the modulus of the largest eigenvalue) of
the Jacobi matrix is given by
cos(π/M ) + cos(π/N )
2
For the Gauss-Seidel method, we have the following matrix equation
µ=
(D + L)ū(k + 1) = −Uū(k) + ḡ
(124)
(125)
The lower triangular matrix L is on the left hand side, because of the updating procedure discussed
earlier. We can rewrite the procedure as
ū(k + 1) = −(D + L)−1 Uū(k) + (D + L)−1 ḡ
(126)
where −(D + L)−1 U is the Gauss-Seidel matrix.
Suggested exercises
1.1 Derive the finite difference approximations (10), (11) and(12).
1.2 Show that the approximation in 19 is of order O(h) where h is the local mesh.
1.3 Derive the matrix for the 1-D Poisson equation discretized with 3-point nonuniform finite
differences.
24
1.4 Analyze the stability of the finite difference scheme (49) for the diffusion equation
1.5 Use the Fourier method to investigate the stability of the explicit schemes (74) and (75) for the
advection equation.
25