TVE-F 17 005 maj
Examensarbete 15 hp
Juni 2017
A Comparison of Three Time-stepping
Methods for the LLG Equation in
Dynamic Micromagnetics
Anton Kroner
Tomas Berg
Simon Wredh
Abstract
A Comparison of Three Time-stepping Methods for
the LLG Equation in Dynamic Micromagnetics
Anton Kroner & Tomas Berg & Simon Wredh
Teknisk- naturvetenskaplig fakultet
UTH-enheten
Besöksadress:
Ångströmlaboratoriet
Lägerhyddsvägen 1
Hus 4, Plan 0
Postadress:
Box 536
751 21 Uppsala
Telefon:
018 – 471 30 03
Telefax:
018 – 471 30 00
Hemsida:
http://www.teknat.uu.se/student
Micromagnetism is the study of magnetic materials on the microscopic length scale (of
nano to micrometers), this scale does not take quantum mechanical effects into
account, but is small enough to neglect certain macroscopic effects of magnetism in a
material.
The Landau-Lifshitz-Gilbert (LLG) equation is used within micromagnetism to
determine the time evolution of the magnetisation vector field in a ferromagnetic
solid. It is a partial differential equation with high non linearity, which makes it very
difficult to solve analytically. Thus numerical methods have been developed for
approximating the solution using computers.
In this report we compare the performance of three different numerical methods for
the LLG equation, the implicit midpoint method (IMP), the midpoint with
extrapolation method (MPE), and the Gauss-Seidel Projection method (GSPM).
It was found that all methods have convergence rates as expected; second order for
IMP and MPE, and first order for GSPM. Energy conserving properties of the schemes
were analysed and neither MPE or GSPM conserve energy. The computational time
required for each method was determined to be very large for the IMP method in
comparison to the other two. Suggestions for different areas of use for each method
are provided.
Handledare: Doghonay Arjmand & Gunilla Kreiss
Ämnesgranskare: Christian Strietzel
Examinator: Martin Sjödin
ISSN: 1401-5757, TVE-F 17 005 maj
Populärvetenskaplig sammanfattning
Mikromagnetism är ett område som studerar magnetism på mikroskopiska (dock ej kvantmekaniska) längdskalor. Området är av intresse bland annat inom hårddiskindustrin, där
magnetiska material används för att lagra information.
Inom mikromagnetism finns Landau-Lifshitz-Gilberts (LLG) ekvation. Lösningar
till denna beskriver hur magnetiseringen förändras inom ett material över tid. LLGekvationen är en partiell differentialekvation med variabler i rum och tid, vilket innebär
att lösningen är en funktion som beror på position och tidpunkt.
Ekvationen har en komplicerad struktur och att hitta exakta lösningar är inte lätt.
Följdaktligen använder man numeriska metoder och datorer för att beräkna ett närmevärde
till lösningen. Det finns ett antal olika metoder som är anpassade för LLG-ekvationen
och syftet med det här projektet är att jämföra egenskaper för tre av dessa: implicita
mittpunktsmetoden (IMP), den extrapolerade mittpunktsmetoden (MPE), samt GaussSiedel projektionsmetoden (GSPM).
Det upptäcktes att alla metoder uppfyllde sina förväntade noggranhetsordningar,
andra ordningen för IMP och MPE, samt första ordningen för GSPM. Energibevaringsförmågan undersöktes och varken GSPM eller MPE klarade av att beräkna energin
korrekt. Beräkningstiden som krävdes för IMP bestämmdes till att vara mycket högre än
för de andra två metoderna. GSPM är lämpig för snabba och onoggranna simuleringar.
MPE bör användas för beräkningar som inte kräver exakt energibevaring och som uppfyller dess stabilitetsvillkor. IMP ger mest korrekta lösningar oavsett diskretisering men
är relativt långsam.
Contents
1 Introduction
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
1
1
2 Theory
2.1 The LLG equation . . . . . . . . . . . . . .
2.2 Energy . . . . . . . . . . . . . . . . . . . . .
2.3 Numerical properties . . . . . . . . . . . . .
2.4 Numerical methods . . . . . . . . . . . . . .
2.4.1 Implicit midpoint method . . . . . .
2.4.2 Midpoint with explicit extrapolation
2.4.3 Gauss-Seidel projection . . . . . . . .
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3
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3 Method of comparison
3.1 Physical requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Numerical viability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
11
11
4 Results
4.1 Magnitude conservation
4.2 Energy conservation . .
4.3 Stability analysis . . . .
4.4 Convergence studies . .
4.5 Computation time . . .
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5 Discussion
5.1 Conservative properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Numerical performance . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
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6 Conclusion
19
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1
1.1
Introduction
Background
Magnetism in a material arises from the electron spin and orbital angular momentum
in the atoms of which the material consists of. However, on larger scales, it is possible
to analyse magnetism without taking these quantum mechanical effects into account.
The field of micromagnetics studies magnetism at the microscopic size scale of nano to
micrometers, and has been of interest since the advent of modern computers. In the year
1960, the field had developed enough to be useful in the development of the magnetic
hard drive [1].
A commonly used mathematical model in micromagnetics is the LLG equation. It
was introduced by Landau and Lifshitz in 1935 and later modified by Thomas Gilbert.
The equation is a vectorial partial differential equation (PDE) and is used to describe the
precessional motion of magnetisation in solids, as seen in figure 1. Analytical solutions can
only be found in special cases and consequently numerical schemes have been developed
to reliably solve the equation using computers. [1]
1.2
Purpose
The purpose of this project it to implement three different time stepping methods for the
LLG equation and numerically compare their properties. A classical implicit midpoint
method, known to be a good performer albeit slow is to be put up against two presumably
faster methods: A mid point method with explicit extrapolation and a projection based
scheme implemented with a Gauss-Seidel correction. Most importantly the magnitude of
the solution vector has to be conserved for every time step. Furthermore energy conservation is to be studied together with numerical stability, convergence rates and computation
time. These properties are to be weighed against each other to draw conclusions about
efficiency. The methods are to be implemented and tested with Matlab.
1.3
Simplifications
Due to the complexity of the equation we have limited ourself to a 1-dimensional setting,
whereas in general the problem is formulated over bounded domains in R3 . The LL
equation, see equation (2.2), includes parameters, which are assumed equal to 1 in all
the simulations here. Moreover, due to the nature of the LL equation, the length of
the magnetisation m remains constant for all the time. Without loss of generality, the
magnitude is also assumed to be 1. As the interest of this project is the time stepping,
we have as much as possible kept to a constant space discretisation of 100 grid points on
the interval [0,100].
1
2
2.1
Theory
The LLG equation
The Landau-Lifshitz-Gilbert-equation is used to determine the time-evolution of the magnetisation field in a ferromagnetic material (for a derivation see: [2]). To simplify equations this paper will only consider its normalised form which can be written as
∂m
∂m
= − m × hef f − am ×
,
(2.1)
∂t
∂t
where m(x, t) is the normalised magnetisation vector field of the ferromagnetic material,
hef f is an effective magnetisation field and a is a damping constant. The equation can
also be written on the equivalent [3] Landau-Lifshitz (LL) form
∂m
= −βm × hef f − αm × (m × hef f ),
∂t
(2.2)
which is explicit with respect to the time derivative of the magnetisation field. In this
formulation β is a gyromagnetic constant and α is a damping constant. The LL version
is suitable for explicit numerical schemes whereas the LLG variant can be beneficial for
implicit schemes [4]. The effective magnetisation field is given by
hef f =
2A
∇2 m + hani + hext + hd ,
2
µ0 Ms
(2.3)
where A is an exchange constant, µ0 is the permeability and Ms is the magnitude of
the unnormalised magnetisation field. hani is the anisotropic magnetisation field, hext is
the external magnetisation field and hd is the demagnetisation field. To calculate hd we
would need to couple the LLG equation with one of Maxwell’s equations, however in this
study we only want to study methods for the LLG equation so the demagnetisation field
has been set to zero.
This study is limited to one space dimension which reduces the laplacian to a single
second order derivative. If we have uniaxial anisotropy, along a unit vector, p, we can
write the anisotropy field as [5]
hani = hm, pip
(2.4)
and as the external field is any known applied field, we can write the effective field as
hef f = B
∂ 2m
+ hm, pip + hext .
∂x2
(2.5)
By taking the inner-product of equation (1.2) with m it is easy to see that |m(t, x)| =
|m(0, x)| for all x. Moreover, it is assumed that |m(0, x)| = 1. This is because the
magnitude of the magnetisation must be equal to the magnetisation saturation of the
material [6]. Figure 1 shows the precessional motion of a single magnetisation vector
under the effect of an effective h-field.
2
z
b
a
Heff
y
m
x
Figure 1: The path of the magnetisation vector, m, as it precesses around the effective H
field, where a and b are the first and second terms in the LLG equation, respectively [4].
2.2
Energy
Magnetic materials exhibit very complex behaviour due to the competition between different energy terms. There are several contributions describing different phenomena that
add up to a total free energy [4]. In this report we are considering three of these contributions.
The exchange energy is given by
Z
1
Jexc |∇m|2 ,
(2.6)
Eexc =
2
where Jexc is a material constant.
The anisotropy energy is given by
Eani
1
=−
2
Z
Kani hm, pi2 ,
(2.7)
where Kani is a material constant and p is the direction of an easy axis in a uniaxial
material.
The applied field energy is given by
Z
Eext = − µ0 hhext , mi,
(2.8)
where hext is an externally applied field. Important to note is that when the damping
term vanishes the LL equation (2.2) conserves energy [4]. This conservative property is
typically sought after when designing a numerical method for approximating the solution.
With N space steps the total continuous energy can be discretised as follows: [7]
N
N
N
X
1X
1X
E=−
Jexc (mk · mk−1 + mk · mk+1 ) −
Kani (mk · p)2 −
µ0 hext · mk (2.9)
2 k
2 k
k
3
2.3
Numerical properties
In numerical analysis there are certain core properties that play vital roles for the viability
of a method. First of all, an explicit method calculates the solution using only current and
previous time states. Comparatively a method of implicit nature expresses the solution
at the upcoming time state in terms of the solution itself. Consequently an equation
needs to be solved for every time step when using an implicit scheme.
The notion of stability, in general, is used to describe the sensitivity of the solution
when the data in a given model problem are perturbed slightly. A necessary condition for
the convergence of a numerical solution towards the exact solution is the stability of the
numerical method, which may be fulfilled under some constraints on the discretisation
parameters, as is the case for most explicit methods. Nevertheless, there are methods
(implicit methods) which are unconditionally stable. Naturally, for a conditionally stable
scheme to be usable at all the discretisation constraint has to be fulfilled.
To determine how accurate the solution of a method is one can analyse the order of
convergence/accuracy. This is a number p that gives information about how the error relates to a discretisation ∆t as shown in equation (2.10).
∝ ∆tp
(2.10)
Different methods use different algorithms that require certain amount of computing
power. For large problems or small discretisation parameters the computational time can
increase to unfeasible scales. Although dependent on hardware and code optimisations,
a comparison can be achieved by simply timing the execution time.
2.4
Numerical methods
The nonlinearity of the LL and the LLG equations makes it very difficult to determine
a solution analytically, this results in the use of numerical methods to reliably solve the
equations. The part that gives rise to interesting problems is the time discretisation and
there are different numerical methods that can be used to solve the equations, with their
own strengths and weaknesses. This report will focus on analysing the properties of three
specific methods.
2.4.1
Implicit midpoint method
Finite difference methods work by replacing the derivatives with finite differences that
approximate the derivative at a discrete point,
y n+1 − y n
dy(tn+1 )
≈
dt
∆t
(2.11)
where n denotes the time step and y n is an approximation to the solution at y(tn ). The
differential equation then reduces to a series of simple algebraic equations that can be
solved by ”stepping” as long as y 0 is known.
The popular Crank-Nicolson method uses an average of the Euler forward and Euler
backward methods to achieve second order accuracy. It may also be referred to as the
4
implicit midpoint (IMP) scheme and can be written on the form
1
y n+1 − y n
= f (tn+ 1 , y n+ 2 ),
2
∆t
(2.12)
where
1
1
(2.13)
y n+ 2 = (y n+1 + y n ).
2
Implicit means that unlike equation (2.11), it can not be directly solved because y n+1
depends on itself.
However, methods for solving non linear algebraic equations have been known for a
long time, and the Newton-Rhapson method is commonly applied for solving problems
of this type. Say we have some function g(x), the Newton-Rhapson method can is used
to approximate a root, x ,that solves this equation by iteration given a starting ”guess”
x0
g(xi )
xi+1 = xi − 0
(2.14)
g (xi )
where g 0 (x) is the derivative of g(x). The stopping criterion for the algorithm is kxi+1 − xi k <
tol for a given value of tol.
We can apply equation (2.14) to solve (2.12) by rearranging it and giving it a new
name, G(y n+1 ), we would then iterate the solution y n+1 in each time step, and use y0n+1 =
y n , the solution in the previous time step, as a starting guess.
Even though the LL equation is a vector PDE, the method works exactly the same.
We discretise in time by replacing the time derivative in equation (2.2) with a finite
difference
mn+1 − mn
∂m(tn )
≈
(2.15)
∂t
∆t
and using the midpoint average, equation (2.13), on the right hand side we get
1
1
1
mn+1 − mn
= −βmn+ 2 × hef f − γmn+ 2 × (mn+ 2 × hef f ).
∆t
(2.16)
Now define G(mn+1 ) by rearranging equation (2.16) into
G(mn+1 ) =
1
1
1
mn+1 − mn
+ βmn+ 2 × hef f + γmn+ 2 × (mn+ 2 × hef f ).
∆t
(2.17)
Now we want to use Newton-Rhapsons method to solve for mn+1 , however, both G and
mn+1 are vectors. Equation (2.14) can be expanded into solving systems of equations by
replacing the scalar quantities by their respective vector equivalents. The derivative in
equation (2.14) is replaced by the Jacobian matrix of G, J, which gives us
n+1
mn+1
− J−1 G(mn+1
)
i+1 = mi
i
(2.18)
where as before we use mn+1
= mn . Notice that at this point, it is important to not
0
confuse the Newton-Rhapson iteration index, i, with the time stepping index, n. The
n+1
time stepping index increases only when the kmn+1
k < tol condition is met, and
i+1 − mi
n+1
mi ”becomes” the solution for that time step after a series of iterations of over i. That
value is then used as an initial guess for calculating the solution at the next time step.
5
To calculate hef f we must now choose a discretisation in space. Even a one-dimensional
domain has three components of the vector m at each point in space, so it is useful to
represent the magnetization field by a N × 3 matrix, where N is the number of discrete
space points, and the columns contain the components of the vector at each point in
space.


mx0
my0
m z0
 mx
my1
m z1 
1


 mx

m
m
y2
z2 
2

m =  ..
..
.. 
 .
.
. 


mxN −1
myN −1
mzN −1 
mxN
myN
mzN
Figure 2: Space discretisation matrix, where mx,y,z denotes a component of m
We replace the derivative in equation (2.5) by a finite difference approximation
mk−1 − 2mk + mk+1
∂ 2 m(xk )
≈
2
∂x
∆x2
(2.19)
where k ∈ {0, 1, 2, . . . , N } denotes a point in space. By expanding equation (2.19) into
the component form

 2
∂ mx (xk )
2


 ∂x 
mxk−1 − 2mxk + mxk+1


1 
 ∂ 2 my (xk ) 
myk−1 − 2myk + myk+1 
(2.20)
 ∂x2  ≈
 ∆x2

m
−
2m
+
m


zk−1
zk
zk+1
∂ 2 mz (xk )
∂x2
we can see that this discrete derivative can be represented by a multiplication of each
column of the space discretisation matrix, m, by a matrix


−2 1
 1 −2 1


1 
..


.
1 −2
A=
(2.21)


2

∆x 
.
.
.. .. 1 

1 −2 N ×N
containing the coefficients of the finite difference approximation. This leads to
∂ 2m
≈ Am
∂x2
(2.22)
which is the second order space derivative. However, equation (2.22) is not complete
without providing boundary conditions. One type of boundary conditions are the periodic
6
boundary conditions, which means that we imagine the domain as a circle. To implement
this, we modify A to contain two additional non zero elements in the corners


−2 1
1
 1 −2 1



1 
..

.
1 −2
A=
(2.23)


2

∆x 
.. ..


.
. 1
1
1 −2 N ×N
which will make it so that mx,y,zN +1 = mx,y,z0 and mx,y,z−1 = mx,y,zN , completing equation
(2.22).
We can now define a discrete version of the effective field, hef f , from equation (2.5).
In place of the derivative we use equation (2.22) and the other terms follow from their
discrete counterparts and we can write
hef f = BAm + hm, pip + hext
(2.24)
where p, hext and hef f have been extended to be matrices that contain their respective
vector components at a point n at each row, analogous to m (see: figure 2.4.1). The
inner product is taken at each individual row of m and p to return a scalar that scales
each row of p.
Finally, we can combine both the time and space discretisations. We extend equation
(2.17) by replacing each m with m and hef f by hef f
1
1
1
mn+1 − mn
+ βmn+ 2 × hef f + γmn+ 2 × (mn+ 2 × hef f )
∆t
where G has been extended to a matrix, as before. And
G(mn+1 ) =
1
1
hef f = BAmn+ 2 + hmn+ 2 , pip + hext
(2.25)
(2.26)
uses the midpoint average
1
1
mn+ 2 = (mn+1 + mn )
(2.27)
2
from equation (2.13) just like previously.
G is now three systems of N equations, but equation (2.18) still applies, except that
for each iteration we have to compute a Jacobian for each column of G, multiply them
together separately, and finally put them back into a matrix together. We write this as
n+1
n+1
n+1
n+1
) J−1
) J−1
)
(2.28)
mn+1
− J−1
y Gy (mi
z Gz (mi
x Gx (mi
i+1 = mi
where, just as before, we use mn+1
= mn as an initial value, Jx,y,z are the Jacobians
0
of Gx,y,z , which are the columns of G, and i is the iteration index that runs until the
n+1
tolerance kmn+1
k < tol is fulfilled.
i+1 − mi
The Jacobians in each iteration are calculated by a central finite difference
Gkx (mkx + d2 , mky , mkz ) − Gkx (mkx − d2 , mky , mkz )
=
d
k
d
k
k
k
Gy (mx , my + 2 , mz ) − Gky (mkx , mky − d2 , mkz )
k
Jy =
d
k
d
k
k
k
Gz (mx , my , mz + 2 ) − Gkz (mkx , mky , mkz − d2 )
k
Jz =
d
Jkx
7
(2.29)
where k runs over every row of the spatial discretisation matrix, and d is a small number
(≈ 10−6 ) used to compute the finite difference.
2.4.2
Midpoint with explicit extrapolation
The midpoint with explicit extrapolation (MPE) uses, just like Runge-Kutta methods for
ordinary differential equations, a weighted average of several different time-steps when
calculating the next time-step [4].
If one considers the LL equation (2.2), we see that it can be written on the following
form
∂m
= −γm × H,
(2.30)
∂t
where
H = hef f + αm × hef f ,
(2.31)
where α is a damping constant. We can see in equation 2.30 that m · ∂m
= 0 and as a
∂t
consequence that kmk = const [8].
To create a numerical method we introduce the following midpoint approximation
dm
dt
n+ 21
=
mn+1 − mn
+ O(∆t2 ),
∆t
(2.32)
mn+1 + mn
+ O(∆t2 ),
(2.33)
2
where the subscript n indicates the time-step. The O(∆t2 ) is used to denote the order of
accuracy in the time region, this being a order of accuracy of 2.
To derive a numerical method we also need to express H in the midpoint, we can do
this by extrapolating H as follows
1
mn+ 2 =
1
1
Hn+ 2 = hn+ 2 + O(∆t2 ),
(2.34)
where
1
1
3
(2.35)
hn+ 2 = Hn + Hn−1 .
2
2
We can now substitute equations 2.32, 2.33 and 2.34 into the LL equation 2.30. This
gives us the following expression
mn+1 − mn =
1
γ∆t n+1
(m
+ mn ) × hn+ 2 ,
2
(2.36)
which can be expressed as
mn+1 +
1
1
γ∆t n+1
γ∆t n
m
× hn+ 2 = mn −
m × hn+ 2 .
2
2
8
(2.37)
This gives rise to a system of equations that can be expressed as a product of matrices


1
1
γ∆t n+ 2
γ∆t n+ 2  n+1 
h
h
1
−
z
y
mx
2
2

1 
γ∆t n+ 2  mn+1 
− γ∆t hn+ 21
1
hx
y
 2 z1

2
1
mn+1
γ∆t n+ 2
γ∆t n+ 2
z
hy
− 2 hx
1
2


1
1
γ∆t
n n+ 2
n n+ 2
n
m − 2 (my hz − mz hy )
 x
1
1 

2 ,
= mny − γ∆t (mnz hxn+ 2 − mnx hn+
)
z
mnz
−
2
1
γ∆t
n n+ 2
(m
h
y
x
2
n+ 12
− mny hx
)
where the subscripts x,y and z represents the components of the corresponding vectors.
Now that we have a numerical method for temporal discretisation we can take a look
at the spatial discretisation. The spatial discretisation lies in the effective field hef f or
∇2 m. To descretizise this we use a second order
more precisely the exchange field µ02A
Ms2
finite difference scheme with an order of accuracy of 2:
∇2 m =
∂ 2m
mk+1 − 2mk + mk−1
=
+ O(∆x2 ).
2
2
∂x
∆x
(2.38)
This is then simply substituted into the effective field and solved for every k
2.4.3
Gauss-Seidel projection
The Gauss-Seidel projection method (GSPM) applies a Gauss-Seidel correction to a fractional step scheme for symplectic flow of harmonic maps and couples the solution with
a projection method for heat flow of harmonic maps [9]. Considering the undamped LL
equation with only the gyromagnetic term present
∂m
= −m × ∆m,
∂t
(2.39)
it can be linearised using a fractional step scheme [9]
m∗ − mn
= ∆h m∗ ,
∆t
mn+1 = mn − mn × m∗ ,
(2.40)
where ∆t is the time step size and ∆h is the discrete laplacian. Combining the two
equations one gets
mn+1 = mn − mn × (I − ∆t∆h )−1 mn .
(2.41)
This scheme can be shown to be unstable when applied to a PDE such as the LLG
equation [10]. Due to its vectorial product structure the stability of the scheme can be
significantly improved by implementing an implicit Gauss-Seidel correction where new
component values are reused immediately. In its component form the improved scheme
looks as follows
gn = (I − ∆t∆h )−1 mn
(2.42)
 n+1  

n
n n
n n
mx + (gy mz − gz my )
mx
n
n n+1
mn+1


m
− gxn+1 mnz )  .
=
(2.43)
y
y + (gz mx
n+1
n
n+1 n+1
n+1 n+1
mz
mz + (gx my − gy mx )
9
When the gyromagnetic term is omitted and we have a case of infinite damping the
LL equation becomes
∂m
= −m × (m × ∆m),
(2.44)
∂t
or equivalently
∂m
= ∆m + ∇|m|2 m.
(2.45)
∂t
(2.45) describes heat flow for harmonic maps. An unconditionally stable two-step projection scheme for the equation is shown in [11]. The term ∇|m|2 m is considered a point
wise constraint: hm,mi = 1. First solve
m∗ − mn
= ∆h m∗
∆t
for m∗ , then project onto the unit sphere
m∗
.
mn+1 =
km∗ k
(2.46)
(2.47)
The two above methods for solving equations (2.39) and (2.45) can be coupled together
to solve the full LL equation as the Gauss-Seidel projection method [10]. Considering a
splitting of the effective field
hef f = ∆m + f,
(2.48)
where
f = hani + hext ,
(2.49)
the full equation can be written as
∂m
= −βm × (∆m + f) − αm × m × (∆m + f).
∂t
GSPM can be applied to (2.50) in three steps:
Step 1: Implicit Gauss-Seidel for gyromagnetic term
gn = (I − ∆t∆h )−1 (mn − f)
g∗ = (I − ∆t∆h )−1 (m∗ − f)

 ∗  ∗
mx + (gyn mnz − gzn mny )
mx
m∗y  =  m∗y + (gzn m∗x − gx∗ mnz )  .
m∗z + (gx∗ m∗y − gy∗ m∗x )
m∗z
(2.50)
(2.51)
(2.52)
Step 2: Projection method for heat flow of harmonic maps
f∗ = h∗ani + hext
 ∗∗   ∗

mx
mx + α∆t(∆h m∗x ∗ +fx∗ )
m∗∗
 = m∗y + α∆t(∆h m∗y ∗ +fy∗ ) .
y
m∗∗
m∗z + α∆t(∆h m∗z ∗ +fz∗ )
z
Step 3: Projection onto unit sphere
 n+1 
 ∗∗ 
mx
m
1  x∗∗ 
mn+1

my .
=
y
km∗∗ k
mn+1
m∗∗
z
z
(2.53)
(2.54)
(2.55)
For the discretisation of the laplacian a second order finite difference scheme was used.
10
3
Method of comparison
To make a comparison several numerical properties and physical constraints have been
investigated for each method. The specific initial value problem we have studied is

∂m

 ∂t = −m × hef f − m × (m × hef f ),
(3.1)
m(0, t) = m(L + ∆x, t),


2πx
2πx
m(x, 0) = [cos( L ), sin( L ), 0],
2
with hef f = ∂∂xm2 + hani + hext , where hani = hm, pip , p = √13 [1, 1, 1] and hext = [0, 0, 1].
For error estimations an ”exact” solution calculated using IMP with ∆t = 10−5 has been
used.
3.1
Physical requirements
Essential from a physical viewpoint is that the magnitude is conserved for every time step,
as explained in section 2.1. The restrictions of this project forces a constant magnitude
of 1. To investigate whether the methods fulfilled this constraint correctly the maximum
norm was calculated for every time step. Its deviation from 1 was then plotted to see the
evolution of the magnitude error.
The gyromagnetic term of the LL equation is of energy conserving Hamiltonian structure [12] . For a numerical method to correctly model the physics this implies that energy
has to be conserved in the case of no damping. The energy conserving properties of the
methods were checked by calculating and plotting the time evolution of the discrete energy as seen in equation (2.9). Although a method does not compute energy correctly it
may still result in a correct solution. This is a consequence of the interaction between
the magnetic moments and the laplacian in the term for the exchange energy.
3.2
Numerical viability
To determine how viable a numerical method is, certain core properties should be analysed, as detailed in section 2.3. To determine which discretisations satisfy correct solutions for a method its stability needs to be analysed. Due to the non linearity of the LL
equation, performing an analytical stability analysis is difficult. Aside for IMP, which is
implicit, we have performed numerical tests where the error is looked at for increasing
time steps with different space steps. If there is a major error increase for a certain discretisation it can be concluded that stability has been lost. Stability results were double
checked by comparing energy values.
How accurate a method is can be concluded from its order of accuracy. This property determines how fast the error decreases with smaller step sizes and is consequently
important for the choice of discretisation. The orders of accuracies were calculated for
each scheme by looking at the errors for different time steps and inserting these values in
equation (2.10).
The last property analysed was the computation time. It is obviously preferable for
a method to be as fast as possible and for large simulations the feasibility might even be
constrained by long execution times. The computation time was investigated by timing
11
the three methods for different amounts of time steps. Linear regression was performed
on the results to get an approximate function for the computation time.
4
Results
General results are compiled in table 1 and a time evolution of the solution can be seen
in figure 3.
Table 1: General results for the three schemes
Scheme Mag. conserv. Energy conserv. Stability Conv. rate Comp. time
IMP
Yes
Yes
Uncond.
2
Long
MPE
Yes
No
Cond.
2
Short
GSPM
Yes
No
Uncond.
1
Short
t = 0.5
1
1
0
0
z
z
t=0
-1
1
0
y
-1
0
50
-1
1
100
0
y
x
-1
1
0
0
z
z
t=2
1
-1
1
y
-1
0
50
-1
1
100
0
y
x
-1
0
0
z
z
1
-1
1
y
0
100
t=5
1
-1
0
50
x
t=3
0
100
x
t=1
0
0
50
50
-1
1
100
0
y
x
-1
0
50
100
x
Figure 3: Solution at different times t. Computed using GSPM with ∆t = 10−4 and 100
grid points on [0,100]. The solution converges towards the effective h-field.
12
4.1
Magnitude conservation
The magnitude conservation was confirmed by finding the largest magnitude error |m|−1
for every time step. Errors for ∆t = 10−3 solutions are plotted in figure 4 and as can be
seen the fluctuations are very small and negligible. The errors for GSPM are on the scale
of machine epsilon.
10 -14
6
Implicit midpoint
4
2
0
0
1000
1500
10 -14
2
|m|-1
500
2000
2500
3000
3500
4000
4500
5000
3500
4000
4500
5000
4000
4500
5000
Midpoint extrapolation
1
0
0
500
1000
10 -16
3
1500
2000
2500
3000
Gauss-Seidel projection
2
1
0
0
500
1000
1500
2000
2500
3000
3500
Time steps
Figure 4: Magnitude fluctuations around 1 for the three methods with ∆t = 10−3 and
100 grid points on [0,100]
4.2
Energy conservation
With zero damping the equation reduces to
∂m
= −m × hef f ,
∂t
(4.1)
which is of Hamiltonian structure and conservative with respect to energy. To observe
how well the methods conserve this property their discrete energies were calculated using
13
equation (2.9). The energy evolution over time for the three methods are shown in figure
5. It is clear from the plots that implicit midpoint conserves energy whereas the other
two schemes do not. However, the projection method diverges a lot faster than midpoint
with extrapolation which is very clear from figure 6 that shows the energies in the same
plot.
Implicit midpoint
-116.3
-116.4
-116.5
0
10
20
30
50
60
70
80
90
100
70
80
90
100
70
80
90
100
Midpoint extrapolation
-116.3
Energy
40
-116.4
-116.5
0
10
20
30
40
50
60
Gauss-Seidel projection
-100
-120
-140
-160
-180
0
10
20
30
40
50
60
Time
Figure 5: Energy evolution without damping for the three methods with ∆t = 10−2 ,
T = 100 and 100 grid points on [0,100].
14
Energy evolution without damping
-116
-116.5
Energy
-117
Implicit midpoint
Midpoint extrapolation
Gauss-Seidel projection
-117.5
-118
-118.5
-119
0
20
40
60
80
100
Time
Figure 6: Energy evolution without damping for the three methods on the same energy
scale with ∆t = 10−2 , T = 100 and 100 grid points on [0,100].
4.3
Stability analysis
Due to the complexity of the LL equation performing analytical stability analyses is
rather difficult. Instead one can analyse the methods numerically by observing how the
error behaves at different time steps. If the error suddenly spikes and starts to behave
erratically the stability criteria is no longer fulfilled and the method has become unstable.
The IMP method is of implicit nature and therefore unconditionally stable. According
to [9] the GSP method is also unconditionally stable. Test runs with large time steps
have not been able to disprove this thesis and it is thus assumed true.
MPE is an explicit scheme and should lose stability for large step sizes. Figure 7 shows
three plots for different space steps around time steps where there is a marginal increase
in maximum error. The error was approximated by comparing the last time step with
solutions of finer discretisations. From this information an approximation of the stability
relation was calculated. The plots in figure 7 implies a stability relation that follows the
∆t
trend ∆x
2 ≤ C, where C is some constant. From table 2 it is possible to approximate the
value of the constant C to: 0.1539 ± 3.22%.
This result can also be confirmed by analysing the energy evolution with damping
around time steps close to the stability limit. As shown in figure 8 the energy starts
15
increasing after supposed convergence for time steps larger than the stability condition.
Table 2: ∆t for loss of stability for different ∆x
∆t
∆x
0.143
1
0.087 0.75
0.041 0.5
Stability dx=1
1
0.5
0
0.02
0.04
0.06
0.08
0.12
0.14
0.16
0.18
0.2
0.14
0.16
0.18
0.2
0.14
0.16
0.18
0.2
Stability dx=0.75
2
ERROR
0.1
1
0
0.02
0.04
0.06
0.08
0.1
0.12
Stability dx=0.5
4
2
0
0.02
0.04
0.06
0.08
0.1
0.12
Time Step
Figure 7: MPE: Error as a function of ∆t for three different space steps.
16
Energy evolution with damping for different
-100
t
-120
t = 0.143
t = 0.149
t = 0.159
t = 0.189
Energy
-140
-160
-180
-200
-220
-240
0
2
4
6
8
10
Time
Figure 8: MPE: Energy over time with damping for different time steps with 100 grid
points on [0,100].
4.4
Convergence studies
Convergence rates were tested by using an IMP solution of (3.1) with ∆t = 10−5 and 100
grid points on [0,100] as ”exact”. Absolute errors for decreasing time steps of each of the
methods were then computed. The errors were calculated as the maximum error norms
of all space steps at the end time T = 5. These results confirm an order of accuracy of 2
for IMP and MPE and 1 for GSPM.
Table 3: Errors
∆t
0.1
0.1/2
0.1/4
0.1/8
0.1/16
Order of accuracy
at time T = 5 for different ∆t
IMP
MPE
GSPM
2.9842E-4 1.0808E-3 8.7727E-3
7.4119E-5 2.6964E-4 4.0920E-3
1.8570E-5 6.7320E-5 1.9713E-3
4.7222E-6 1.6996E-5 9.6701E-4
1.2874E-6 4.4605E-6 4.7894E-4
1.9862
1.9829
1.0471
17
4.5
Computation time
A major difference between the IMP method and the other methods is the computational
time required to complete a time step. Computational time was measured for each of the
methods to find the dependency on the amount of time steps. In table 4 we can see that
the GSPM is the fastest, with a computational time of about 2 seconds per 1000 time
steps. The MPE method is about half as fast, but the IMP method has a time cost that
is more than two orders of magnitude greater than the GSPM.
Table 4: Computation time, in seconds, for each method using K time steps with ∆x = 1.
K
IMP
MPE
GSPM
100
18.43
0.59
0.29
400
60.96
1.54
0.75
800
105.17
2.93
1.42
1000
132.17
3.54
1.76
4000
423.34
13.67
7.36
8000
834.59
27.33
14.54
10000
1017.66
34.09
18.38
Tcomp ≈ 0.1K + 22 0.0034K + 0.2 0.0018K
5
5.1
Discussion
Conservative properties
We know from the theory that kmk is supposed to be constant over time. Looking at
figure 4 that shows how the magnitude error evolves we can see that all three methods
conserve this property well. The errors are very small and should be negligible in almost
all cases. The IMP plot has a rather odd increase but the error is still very small and
in a run with more time steps the error stayed at the same order of magnitude, similar
to the MPE case. For GSPM m is normalised in every time step and consequently the
error is constantly on the scale of machine epsilon.
Figure 6 shows the energy over time and we can see that the IMP method conserves
energy very well. This agrees with [13] which shows analytically that the method should
conserve energy. Furthermore it is clear that both GSPM and MPE fails to keep the
energy constant. Although, whereas the projection method quickly diverges the midpoint
scheme does fairly well with a relatively low margin of error. In situations where the
energy quantity is of importance the GSPM is consequently not ideal to use but MPE
might be suitable in some cases.
5.2
Numerical performance
IMP is an implicit scheme and therefore unconditionally stable. The lack of stability
constraint is one of the main benefits of this method. Combined with an order of convergence of two and its energy conserving properties the IMP method is a popular LLG
solver. However, due to the implicit nature its downsides are implementation difficulties
18
as well as long computation times. (Although not considered in this project, it should
be possible to reduce the computational time of the IMP method by disregarding a few
elements when computing the approximation to the Jacobian matrix [13].)
Comparatively MPE is explicit and as figure 7 shows stability is lost for large time
steps in relation to the space step. The constraint for the studied problem was approximated to be ∆t 6 0.154∆x2 which allows for some discretisation variety but not as
freely as with IMP. As the order of convergence also is two what differs the most from
IMP is computation time where MPE performs a lot better. In cases where the stability
constraint is not limiting and energy does not have to be very accurate MPE seems to
be the preferred mid point scheme.
From table 4 we can see that GSPM is the fastest scheme, although similar to MPE.
Seemingly unconditionally stable as well, it is suited for quick computations. However,
it is unable to conserve energy and only has an order of convergence of one. It is thus
debatable whether GSPM can hold up to the two mid point methods when more accurate
solutions are required.
6
Conclusion
We implemented three different methods for solving the LLG equation. Experiments to
confirm convergences rates were performed and the results match expected rates from
cited sources. The energy conservative properties of the solutions to the undamped LLG
equation were investigated and it was found that the GSPM and the MPE method failed
to compute this correctly. The stability constraint for the MPE method was approximated. The computational time for the IMP method was found to be two orders of
magnitude greater than the other two methods.
We can therefore conclude that the IMP should be used in cases where energy computation, or the use of large time steps is of importance. The GSPM should be used for
fast calculations where the energy is of negligible importance. And the MPE could be
used in cases where some energy inaccuracy is allowed and necessary discretisations fulfil
the stability criterion as a faster replacement for the IMP.
19
References
[1] D. Wei, Micromagnetics and recording materials. Springer Science & Business Media,
2012.
[2] B. Gurney, M. Carey, C. Tsang, M. Williams, S. Parkin, R. Fontana Jr, E. Grochowski, M. Pinarbasi, T. Lin, and D. Mauri, “Ultrathin magnetic structures iv,
edited by b. heinrich and jac bland,” 2005.
[3] G. Bertotti, Hysteresis in magnetism: for physicists, materials scientists, and engineers. Academic press, 1998.
[4] I. Cimrák, “A survey on the numerics and computations for the landau-lifshitz
equation of micromagnetism,” Archives of Computational Methods in Engineering,
vol. 15, no. 3, pp. 1–37, 2007.
[5] L. Baňas, “Numerical methods for the landau-lifshitz-gilbert equation,” Numerical
Analysis and Its Applications, pp. 158–165, 2005.
[6] A. Aharoni, Introduction to the Theory of Ferromagnetism, vol. 109. Clarendon
Press, 2000.
[7] D. Arjmand, M. Poluektov, and G. Kreiss, “Atomistic-continuum multiscale
modelling of magnetisation dynamics at non-zero temperature,” arXiv preprint
arXiv:1702.05173, 2017.
[8] C. Serpico, I. Mayergoyz, and G. Bertotti, “Numerical technique for integration of
the landau–lifshitz equation,” Journal of Applied Physics, vol. 89, no. 11, pp. 6991–
6993, 2001.
[9] X.-P. Wang, C. J. Garcıa-Cervera, and E. Weinan, “A gauss–seidel projection method for micromagnetics simulations,” Journal of Computational Physics,
vol. 171, no. 1, pp. 357–372, 2001.
[10] C. J. Garcia-Cervera, “Numerical micromagnetics: A review,” 2007.
[11] E. Weinan and X.-P. Wang, “Numerical methods for the landau–lifshitz equation,”
SIAM journal on numerical analysis, 2006.
[12] E. Kim and K. Lipnikov, “The mimetic finite difference method for the landau–
lifshitz equation,” Journal of Computational Physics, vol. 328, pp. 109–130, 2017.
[13] M. d’Aquino, C. Serpico, and G. Miano, “Geometrical integration of landau–lifshitz–
gilbert equation based on the mid-point rule,” Journal of Computational Physics,
vol. 209, no. 2, pp. 730–753, 2005.
20