Equation Modification Methods
Carlos A. Felippa
Department of Aerospace Engineering Sciences
and Center for Aerospace Structures
University of Colorado, Boulder
Boulder, CO 80309-0429, USA
April 2001
We modify the solution of the former problem without changing the solution of the latter. (R. Courant)
A rose by any other name is just as sweet. (W. Shakespeare)
1 INTRODUCTION
The purpose of this note is to discuss a question of ad-hoc terminology used by researchers involved in
numerical methods. For example, FEM workers talk about “stabilization techniques”, “dynamic relaxation”,
“residual bubbles” and “subgrid methods”; CFD people talk about “modified differential equations” and
“defect corrections”; numerical analysts talk about “embedding” and “homotopies.”
Some variants and outgrowths are sufficiently different and promisory to merit new names.1 However, most
often than not they are just variations on a theme sharing a common mathematical framework that can be
used to advantage.
Modification
(augmentation,
embedding)
Original
equation
r(u)=0
Modified equation
r(u)+s(u,h)=0
Make h=0
Figure 1. Equation modification concept.
2 THE COMMON FRAMEWORK
The basic idea shared by all these methods is sketched in Figure 1. For definiteness, assume you want to solve
an equation r (u) = 0, for u. This is called the residual form. The unknown u may be any mathematical
object: variable, vector, matrix, function, etc., with the meaning of r and the zero on the RHS adjusted
accordingly.
Suppose that for some reason(s) it is convenient to transform the problem as shown in Figure 1. The equation
r (u) = 0 is replaced by a modified equation r (u) + s(u, h) = 0.2 The extra term, which as noted receives
many names in the literature, depends on some parameters collected in h as well as on u. Although h may
be given many interpretations, it is often a set of scalar variables (that is, a vector h = [h i ] which is chosen
1
The finite element method (FEM) is an example of apt renaming. Although its mathematical underpinnings follow classical
Ritz-Galerkin, the method embodies far more than mathematics and hence deserves its own name.
2
Instead of this strong form, the original and modified problem could be posed ab initio in weak or variational form. Indeed Courant’s
formulation discussed in §5.3 is based on functionals discretized by the classical Rayleigh-Ritz method. The strong form is used in
the exposition as it is more general and compact.
1
so that if h i → 0, s(u, h i ) → 0) and the modified equation reduces to r (u) = 0. In physical applications,
the h i can be given some physical interpretation as associated with lengths or time dimensions.3
What is “some reason”? The most common objectives are:
1.
To improve a given solution method in terms of robustness, stability or performance.
2.
To facilitate the analysis, as well as general understanding, of a specific solution method.
3.
To develop new solution methods.
4.
To link up existing and new solution methods.
5.
To incorporate multiscale physics, e.g. boundary layers or subgrid effects, in the modification terms.
The last reason is one receiving much recent attention from the FEM community.
2.1 Modify, Embed or Augment?
The terms modification, embedding and augmentation shown in Figure 1 deserve comment.
Modification is the more neutral term and is the one preferred here. It connotes incremental and reversible
change; that is, one can recover the original problem through the simple device of making h (or the h i ) zero.
But modifications should not be interpreted in the sense of small perturbations; the latter are a restricted,
and not very useful specialization, of the former.
Embedding is more grandiose: make a problem a special case of a wider one. For example, mapping a
steady state problem into a dynamical one whose long-time behavior reproduces that steady state.
Augmentation is more specific: to transform a problem or pieces of a problem by adjoining mathematical
objects such as new terms, additional variables or matrix bordering. The new problem may be equivalent to
the original one, or belong to a wider class; in the latter case we speak of embedding by augmentation.
3 WHERE DOES THE SOLUTION METHOD FIT IN?
Figure 1 is devoid of practical details. In particular, it does not show where the discretization-solution method
fits in the picture. After all, remember that we are interested in solving r (u) = 0, at least approximately.
We discuss below two scenarios.
3.1 Scenario I: Discretize Then Modify
Figure 2 depicts the case in which one selects a method M as candidate to solve the original problem through
some discretization in space and/or time. The natural interpretation for h in this case is some mesh or grid
dimension(s). The discrete solution produced by M is called u h .
By appropriate application of Taylor series to the discretized system a modified equation r (u) + s(u, h)
is constructed. The exact solution of the modified equation reproduces u h . An analysis of the modified
equation may suggest a better discretization method, for example by adjusting free parameters in M. This is
depicted in Figure 2 by the feedback loop into box M.
This particular technique has been used to analyze finite difference methods for hyperbolic systems (especially in CFD) since its introduction by Warming and Hyett.4 Here the original equation typically models
flow effects of conduction and convection, h includes grid dimensions in space and time, and the feedback
3
One feature that distinguishes this idea from classical perturbation analysis is that the h i are at no point assumed to be “small” for
results to be valid. Consequently s(u, h) is not necessarily a perturbation of r .
4
R. F. Warming and B. J. Hyett, The modified equation approach to the stability and accuracy analysis of finite difference methods,
J. Comp. Physics, Vol. 14, (1974), 159–179.
2
M: Numerical discretization
method
Original
equation
r(u)=0
Modified equation
r(u)+s(u,h)=0
Taylor
series
M
Adjust
M
Exact
solution
uh
Figure 2. Discretization method analysis by
Taylor-generated modified equation.
M: Numerical discretization
method
Original
equation
r(u)=0
M
Taylor
series
Fourier
transform
Fourier
transform
Modified
_
_equation
r(u)+s(u,h)=0
Adjust
M
_
uh
Exact
solution
Figure 3. Discretization method analysis by Taylor-generated,
Fourier-transformed modified equation.
is used to adjust M in terms of improving stability as well as reducing spurious oscillations (e.g. by artificial
viscosity or upwinding) and dispersion.
A spectral variant is shown in Figure 3. Upon Taylorizing the discrete system, it is Fourier transformed to
produce a modified equation in transform space ū. This technique was used to analyze locking and spurious
modes in FEM analysis of plates and shells by Flaggs and Park, building on earlier attempts by Strang.5 In
this work the modified equation was called the limit differential equation. The idea is particularly useful for
symbolic analysis of regular discretizations.
3.2 Scenario II: Modify Then Discretize
The second scenario is sketched in Figure 4. The essential change from Scenario I is that the discretization/solution method is applied to the modified equation rather than to the original equation. The final result
is a solution u h or a sequence of solutions u kh .
But a complication appears. To get a modified equation you need to first introduce the h, or related coefficients, in some way. This is done in the box labeled “problem parametrization” in Figure 4. Parametrization
can be done in many ways: a preliminary geometric discretization, definition of characteristic physical
scales, introduction of Lagrange multipliers or penalty weights, etc. What is essential at this point is to have
the h (or h i ) on hand.
The modification is done by a variety of techniques, three of which are listed in Figure 4: Taylor series,
embedding or augmentation. Having obtained the modified equation, the ensuing discretization and solution
5
K. C. Park and D. L. Flaggs, A Fourier analysis of spurious modes and element locking in the finite element method, Comp. Meths.
Appl. Mech. Engrg., 42, 37–46 (1984); K. C. Park and D. L. Flaggs, An operational procedure for the symbolic analysis of the finite
element method, Comp. Meths. Appl. Mech. Engrg., 46, 65–81 (1984).
3
MODIFICATION
Original
equation
r(u)=0
Problem
parametrization
by h
Taylor series,
embedding or
augmentation
Modified equation
r(u)+s(u,h)=0
M: Numerical discretization
method
Solves exactly
r(u)=0 as k→∞
M
Iterative
One pass
u hk
uh
Solves approximately
r(u)=0 as h is reduced
Figure 4. Equation modification followed by discretization and solution.
step has to make some intelligent use of h of the h i . At this point it is useful to divide the solver process into
two types:
One Pass Solver. Gives a solution u h that depends on h. Reducing h in a systematic way (for example,
by a mesh adaptation process) gives a family of approximate solutions that converge to the solution of the
original problem.
Iterative Solver. Gives a solution u kh that depends on h and some index k. As k → ∞ it should reduce to
the solution of the original problem, either independently of the choice of h, or at least for a range of h.
In the second case k plays the role of an iteration index whereas h (or h i ) play the role of method parameters.
Important examples are the solution of nonlinear systems by continuation and/or dynamic relaxation.
4 EXAMPLES OF SCENARIO I
4.1 The Modified PDE Approach of Warming and Hyett
The Introduction of the Warming and Hyett source paper, cited in footnote 4, summarizes their method as
follows.
This paper describes a technique for evaluating various qualities or properties of a finite-difference analogue
of a given partial differential equation. These qualities include order of accuracy, consistency, stability,
dissipation, and dispersion. The technique involves determining the actual partial differential equation which
is solved numerically, aside from round-off error, by the application of a given difference method to solve an
initial-value problem. The actual equation is called the modified equalition. It is derived by expanding each
term of a finite difference equation into a Taylor series and then eliminating time derivatives higher than first
order and mixed and space derivatives by a method described in Section 1. It is emphasized that, contrary to
common practice, the original partial differential equation should not be used to eliminate these derivatives.
Although the modified equation has infinitely many terms, in practice only the first se veral lowest order
terms need to be computed. Terms appearing in the modified equation which are not in the original partial
differential equation represent a type of truncation error. These error terms allow one to determine by
inspection the order of accuracy and consistency of a finite difference approximation.
Although not mentioned in the paper, this method is clearly in the spirit of backward error analysis. The
approximate solution, here that produced by a FD method, is viewed as the exact solution of a neighboring
problem, namely the modified PDE. Backward error analysis has ruled numerical linear algebra since
Wilkinson’s fundamental contributions. But it has not had similar success in the numerical solution of ODEs
and PDEs, which is still dominated by classical truncation error analysis.
4
To show the method in action in an example more related to structural mechanics,6 consider the equation of
motion of a free, undamped, discrete linear mechanical system:
Mü + Ku = 0,
(1)
where M and K are the mass and stiffness matrix, respectively, u = u(t) is the state vector, and superposed
dots denote time differentiation. The mass matrix is assumed nonsingular. We transform (1) to a first order
system using the momentum-displacement version formulated by Felippa and Park in 1978.7 Defining the
momentum p = Mu̇ as auxiliary variable, system (1) becomes Hamiltonian:
u̇ =
dH
,
dp
ṗ = −
dH
,
du
(2)
in which H = 12 pT M−1 p + 12 uKu. In matrix form:
p
ṗ
0
−K
,
=
−1
0
u
u̇
M
or
ż = Az.
(3)
Suppose that (3) is treated by the general one-step implicit LMS integrator
pn+1 = pn + h α ṗn + (1 − α)ṗn ,
un+1 = un + h β ṗn + (1 − β)ṗn .
(4)
Here α > 0 and β > 0 are two free parameters to be determined, subscript n marks time station tn = nh
and h = tn+1 − tn is the time stepsize.
Writing (3) at t = tn+1 and eliminating u̇n+1 andf ṗn+1 from (4) we obtain the algebraic system
1 − α
1
1 I
I
K
0
I
0
ṗn
p
p
α
n+1
n
αh
= αh
+
.
β
−
1
1
1
u̇n
un+1
un
0
I − βh M
0 − βh M
β M
(5)
Next, expand u and p in Taylor series about tn :
...
pn+1 = pn + h ṗn + 12 h 2 p̈n + 16 h 3 p n + . . . ,
...
un+1 = un + h u̇n + 12 h 2 ün + 16 h 3 u n + . . .
(6)
Inserting (6) into (5) and Laplace transforming, the following infinite order ODE results:
(A0 + A1 s + A2 s 2 + A3 s 3 + . . .)z = 0
(7)
˙ ≡ d/dt, and
where s is the Laplace transform of the operator ()
A0 =
h − β1 M
I 0
hI −M
, A1 =
, A2 = 16 h 1
, A3 =
0 K
I hK
I hK
α
1 2
h
24
h − β1 M
, ...
1
I hK
α
(8)
6
A simpler version of this MDE was presented in K. C. Park and C. A. Felippa, Partitioned analysis of coupled systems, Chapter 3
of Computational Methods for Transient Analysis, ed. by T. Belytschko and T. J. R. Hughes, North-Holland, Amsterdam, (1983),
157–219. The analysis given here, which permits different integrators for the momentum and the displacement vectors, is made
possible by the use of Mathematica to carry out the complicated algebraic manipulations.
7
C. A. Felippa and K. C. Park, Computational aspects of time integration procedures in structural dynamics: I. Implementation, J.
Appl. Mech., 45,(1978) 595–602. This paper actually studies the more general case of a damped system, in which there are several
choices for the auxiliary variable.
5
We solve for p and u from (7) in the s domain, expand the resulting expressions in powers of h and return
to the time domain. The result for u can be presented as an infinite quadratic form:


0
0
0
...  
 T K 0 M
1
 0 K 0 a1 M
0
0
... u
 u̇ 
1
h  
0 0 2K
0
a2 M
0
...

 2 


h   0 0 0
 ü  = 0
1
(9)

K
0
a
M
.
.
.
 3 
... 
3
6

h  
u
1

0
K
0
... .
..
0 0 0
24
..
.
.. ..
..
..
..
..
..
.
. .
.
.
.
.
where a1 = (α +β −2α)/(2αβ), a2 = (3−4α −4β +6αβ)/(12αβ), and a3 = (−2+3α +3β −4αβ)/(4αβ).
For the final step, differentiate the first equation repeatedly with respect to time, and use it to eliminate the
diagonal entries in the kernel matrix of (9) except for the first one. The result is the modified differential
equation:


 T K 0 M
0
0
0
...  
1
u
 0 0 0 c1 M
0
0
...
h  
  u̇ 
 2  0 0 0
0
c2 M
0
... 
h  
  ü 
(10)
 3  0 0 0
0
0
c3 M . . .   ...  = 0
 u 
h  
0 0 0
0
0
0
... .
..
..
.. .. ..
..
..
..
..
.
.
. . .
.
.
.
where c1 = (α + β − 4αβ)/(2αβ), c2 = (3 − 10α − 10β + 24αβ)/(12αβ), and c3 = (−8 + 17α + 17β −
32αβ)/(4αβ). In compact form:
...
....
Ku + Mü + hc1 M u + h 2 c2 M u + . . . = 0.
(11)
Evidently the integration method (4) can be made accurate to second order if c1 = 0. This can be obtained
by taking
α + β = 4αβ
(12)
which is a new result. If α = β one obtains α = β = 12 , which is the well known trapezoidal rule (TR). The
TR choice actually minimizes the absolute value of coefficient c2 and gives the MDE:
....
Ku + Mü − 13 h 2 M u + . . . = 0.
(13)
For a one-DOF oscillator system: ü + ω2 u = 0 vibrating at u = A cos ωt + B sin ωt, ü = −ω2 u and
....
....
u = −ω2 ü, whence (13) becomes ω2 u + (1 + 13 ω2 h 2 )ü = 0. Thus the effect of the − 13 h 2 M u term of
the MDE is to increase the inertia thus lowering the frequency and elongating the computed period. This
“equivalent mass” is frequency dependent. Thus if one is integrating a single oscillator equation, a more
accurate TR integration is possible by dividing the mass by 1 + 13 ω2 h 2 .
5 EXAMPLES OF SCENARIO II
5.1 A Linear Homotopy Embedding
Solve
tan x = 1 + x,
or r (x) = tan(x) − 1 − x = 0,
(14)
for the real solution near π. Select x ≈ 1 as initial approximation. Take x = x(h), where h is a continuation
parameter, and embed into the linear homotopy:
H (x, h) = (1 − h)r (x) + h(x − 1) = 0,
H (x, h) = (1 − h)r − r + h (x − 1) + hx = 0.
6
(15)
where prime denotes differentiation with respect to h. Obviously H (1, 1) = 0. Starting from that initial
condition (x = 1, h = 1), integrate H numerically down to h = 0 to get an approximate solution, which
can then be improved with Newton’s method. Here is a Mathematica implementation:
ClearAll[x,h,r,rx];
r[x_]:=Tan[x]-x-1; rx[x_]:=Sec[x]^2-1;
H=Simplify[(1-h)*rx[x[h]]*x’[h]-r[x[h]]+(x[h]-1)+h*x’[h]];
sol=NDSolve[{H==0,x[1]==1},x[h],{h,1,0}]; {xs}=x[h]/.sol/.h->0;
Print["homotopy solution:",xs//InputForm];
Print["residue:",r[xs]];
For[k=1,k<=2,k++, xs=xs-r[xs]/rx[xs];
Print["solution after newton step:",xs//InputForm];
Print["new residue:",r[xs]]];
Running gives:
homotopy solution: 1.1322676105925578
residue: −5.21401 × 10−7
solution after newton step: 1.1322677252729194
new residue: 1.55653 × 10−13
solution after newton step: 1.1322677252728852
new residue: 2.22044 × 10−16
The homotopy solution is correct to 7 digits and two Newton steps raise that accuracy to 16. Although for
this example the introduction of a homotopy is hardly worth the trouble, this incremental-iterative solution
approach is fundamental for more complex nonlinear systems.
The modified equation can be written H/(1 − h) = r + (h/1 − h)(x − 1) = 0 whence the modification
term is s = (h/1 − h)(x − 1). This example fits Scenario II because the modified equation is constructed
first, and then discretized by the numerical integration method.
5.2 Dynamic Relaxation
A FEM-displacement-discretized nonlinear system of equations leads to the residual force equilibrium
equations
r(u, λ) = 0
(16)
where u is the state vector of degrees of freedom, r is the residual vector that contains out-of-balance forces
conjugate to u and λ is a control parameter that scales the magnitude of the applied forces. The problem is
to solve numerically (16) for u at selected values of λ to get the response history u = u(λ).
Suppose a solution (u0 , λ0 ) is known. Parametrize u = u0 + u(t), λ = λ0 + λ(t), where t is a
dimensionless time-like history parameter called the history parameter such that at t = 0 both u and λ
reduce to u0 and λ0 , respectively, and as t → ∞, λ0 + λ(t) → λ∗ at which the next solution u∗ is
desired. Differentiating (16) with respect to t and using superposed dots to denote t derivatives (as in
genuine dynamics) we get
ṙ = Ku̇ − qλ̇ = 0,
r̈ = Kü + K̇u̇ − qλ̈ − q̇λ̇ = 0.
(17)
in which K = ∂r/∂u is the tangent stiffness matrix and q = ∂r/∂λ is the incremental load vector. One
particular dynamic relaxation method takes a linear combination of r, ṙ and r̈ to make a pseudodynamic
system:
r + αh ṙ + βh 2 r̈ = 0.
(18)
7
where h is a stepzise control parameter and (α, β) are adjustable scalars. The idea is to compute the steady
state solution u∗ = u∗ (λ∗ ) of (18) as t is incremented by a stepsize t, related to h, from t = 0 to a
sufficiently large number at which convergence to u∗ is accepted. Many generalizations and of (18) have
been proposed and studied, for example
r + αA(h)ṙ + βB(h)r̈ = 0
(19)
where A and B are weigting matrices that vanish as h → 0. If B = 0 and A = hI one obtains a steplengthcontrolled Newton’s method, sometimes called the damped Newton’s method.
With (19) as our modified system, the added term is s = αA(h)ṙ + βB(h)r̈. This clearly fits Scenario II.
5.3 Courant’s “Sensitized Functionals”
Courant, in a widely cited 1943 article,8 discusses variants of what are now called Courant penalty functions.
One particularly interesting proposal is to make a functional more “sensitive” by adding a higher derivative
term. Here are Courant’s own words:
These facts which are intimately related to more profound questions in the general theory of the variational
calculus have suggested the following method of obtaining better convergence in the Rayleigh-Ritz method.
Instead of considering the simple variational problem for the corresponding boundary value problem, we
modify the former problem without changing the solution of the latter. This is accomplished by adding to
the original variational expression terms of higher order which vanish for the actual solution u. For example,
we may formulate the equilibrium problem for a membrane under the external pressure f as follows:
I (v) =
vx2
+
v 2y
+ v f d xd y +
B
k (v − f )2 d xd y = min .,
B
where k is an arbitrary positive constant or function. Such additional terms make I (v) more sensitive to
variations of v without changing the solution. In other words, minimizing sequences attached to such a
”sensitized” functional will by force behave better as regards convergence.
The practical value of the method of sensitizing the integral by the addition of terms of higher order has
not yet been sufficiently explored. Certainly the sensitizing terms will lead to a more complicated system
of equations for the ci . This means that a compromise must be made for a suitable choice of the arbitrary
positive function k so that good convergence is assured while the necessary labor is kept within bounds.
The Euler-Lagrange PDE of Courant’s sensitized functional is ∇ 2 v + k∇ 4 v = f + k∇ 2 f . Changing the
notation v → u, k → h, and taking r = ∇ 2 u − f this can be written
r + h∇ 2r = 0
(20)
so that s = h∇ 2r = h∇ 4 u. This is clearly a modified equation method that fits Scenario II, because the
paper talks about applying a discretization method (Rayleigh-Ritz) to it.
Courant’s idea drew little attention from the FEM community. The variational index of the modified
functional is raised (in his example, from 1 to 2), which in turn complicates the labor of building elements
with higher continuity. But this disadvantage can be overcome with more advanced tools.
8
R. Courant, Variational methods for the study of problems of equilibrium and vibrations, Bull. Amer. Math. Soc., Vol. 49 (1943),
1–23. This article contains the germs of FEM as a Rayleigh-Ritz method.
8
_
Prescribed
Φ
Primal
BC_
Φ=Φ
Transported
variable
Φ
r(Φ)=0
Transport
velocity
u
Kinematic
g=∇Φ
Gradient
g
Source
Q
Balance
−uTg+∇Tq+Q=0
uTg
Constitutive
Flux
q
q=Dg
(q=kg if isotropic)
Flux BC
_
q=q
Prescribed
_
q
Figure 5. The Strong Form Tonti diagram of the steady-state convection-diffusion problem.
5.4 The Modified Balance Equations of Oñate and Manzan
In a recent report,9 Oñate and Manzan describe a “finite calculus” approach to the construction of improved
FEM discretizations for the convection-diffusion problem. The approach can be classified under Scenario II
because a modified equation is constructed before the FEM discretization. The report treats both the steadystate and transient problems in one and multiple dimensions, but here only the one-dimensional steady-state
case will be summarized. The notation of that report is followed.
Figure 5 summarizes the field equations and boundary conditions of the multidimensional steady-state
convection-diffusion problem, using a Strong Form Tonti diagram. The residual field equation that results
on eliminating the gradient and flux variables10 is
r(Φ) = −uT ∇Φ + ∇ T D ∇Φ + Q = 0
(21)
where Φ is the transported variable, u the transport velocity field viewed as given, ∇ the spatial gradient
operator, and D the diffusivity matrix. For isotropic media the latter reduces to D = k I, where k is the
diffusivity parameter. For one dimension with position coordinate x and uniform isotropic material (21)
specializes to
r (φ) = −uφ + k φ + Q = 0.
(22)
where primes denote differentiation with respect to x. The modified CD equation can be constructed in 1D
as follows. To introduce a characteristic dimension h, consider a space interval of length h extending from
x A = x through x B = x + h. Integrate (22) from A to B for constant k and u to get11
−u(φ B − φ A ) +
k(φ B
−
φ A )
B
+
Q d x = 0,
(23)
A
Expand φ, φ and Q, which are the only x dependent quantities, in Taylor series about A:
1 2 φ B = φ A + hφ A + 12 h 2 φ A +. . . , φ B = φ A + hφ A + 12 h 2 φ A +. . . , Q B = Q A + h Q A + 2 h Q A +. . . (24)
9
E. Oñate and M. Manzan, Stabilization techniques for finite element analysis of convection-diffusion problems, CIMNE Publication
N◦ 183, Feb. 2000.
10
That is, the dashed-line connection between the and Q boxes in Figure 5.
11
Since q = kφ , this equation may be physically interpreted as a flux balance statement: −u(φ B − φ A ) + (q B − q A ) + Q m h = 0,
B
where Q m = (1/ h) A Q d x is the mean source over the interval [A, B]. This is similar to Eq. (54) of the Oñate-Manzan report.
9
Insert these into (23), cancel out one h factor, and drop subscript A. The result is the modified convectiondiffusion equation
1 3 r + 12 hr + 16 h 2 r + 24
h r + . . . = 0.
(25)
where r is the residual defined in (22). The first 3 terms have formally the same configuration as the dynamic
relaxation equation (18) with α = 1/2 and β = 1/6, but the differentiation is here with respect to space, not
pseudotime. In Section 4.1 of the report, Oñate and Manzan consider the first two terms and take h with the
opposite sign:
(26)
r − 12 hr = 0.
and show the equivalence of this form with other well known FEM discretization techniques, such as artificial
diffusion and Petrov-Galerkin. For example, if 12 hr ≈ 12 huφ , which results on neglecting the diffusive and
source terms in r , and h = α, where α is a streamline diffusion coefficient and and element size, one
obtains
r − 12 hr ≈ −uφ + (k + 12 αu)φ + Q = 0,
(27)
which is the augmented residual equation of the artificial streamline diffusion method. If three terms are
retained in (25) in and Q varies superlinearly, the discretization is shown in Section 4.3.1 of the report to be
equivalent to the so-called subgrid-scale model.
The approach is extendible to multiple dimensions, and is useful in establishing the common features of
several existing FEM discretization methods. It could be further combined with the modified PDE method
to extract high order accuracy.
6 CONCLUDING REMARKS
From the wide range of applications and examples cited here, it is obvious that the modified equation
approach provides a common framework that can be used to accomplish a variety of different objectives. To
avoid confusing readers, it is suggested that the method deserves a common name.
7 REFERENCES
An extensive reference list will be compiled in the final version of this draft document. Those consulted are for the
moment in footnotes.
10