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149 PDE 7.2 Partial Differential Equations (PDE)

7.2 Partial Differential Equations (PDE)

PDE overviewExamples of PDE-s:

• Laplace’s equation

– important in many fields of science,

∗ electromagnetism

∗ astronomy

∗ fluid dynamics

– behaviour of electric, gravitational, and fluid potentials

– The general theory of solutions to Laplace’s equation – potential theory

– In the study of heat conduction, the Laplace equation – the steady-stateheat equation

150 PDE 7.2 Partial Differential Equations (PDE)

• Maxwell’s equations – electrical and magnetical fields’ relationships– set of four partial differential equations

– describe the properties of the electric and magnetic fields and relate themto their sources, charge density and current density

• Navier-Stokes equations – fluid dynamics (dependencies between pressure,speed of fluid particles and fluid viscosity)

• Equations of linear elasticity – vibrations in elastic materials with given prop-erties and in case of compression and stretching out

• Schrödinger equations – quantum mechanics – how the quantum state of aphysical system changes in time. It is as central to quantum mechanics as New-ton’s laws are to classical mechanics

• Einstein field equations – set of ten equations in Einstein’s theory of generalrelativity – describe the fundamental interaction of gravitation as a result ofspacetime being curved by matter and energy.

151 PDE 7.3 2nd order PDEs

7.3 2nd order PDEs

We consider now only a single equation caseIn many practical cases, 2nd order PDE-s occur, for example:

• Heat equation: ut = uxx

• Wave equation: utt = uxx

• Laplace’s equation: uxx +uyy = 0.

General second order PDE has the form: (canonical form)

auxx +buxy + cuyy +dux + euy + f u+g = 0.

Assuming not all of a, b and c zero, then depending on discriminant b2−4ac:b2−4ac > 0: hyperbolic equation, typical representative – wave equation;b2−4ac = 0: parabolic equation, typical representative – heat equationb2−4ac < 0: elliptical equation, typical representative – Poisson equation

152 PDE 7.3 2nd order PDEs

• In case of changing coefficient in time, equations can change their type

• In case of equation systems, each equation can be of different type

• Of course, problem can be non-linear or higher order as well

In general,

• Hyperbolic PDE-s describe time-dependent conservative physical processes likewave propagation

• Parabolic PDE-s describe time-dependent dissipative (or scattering) physicalprocesses like diffusion, which move towards some fixed-point

• Elliptic PDE-s describe systems that have reached a fixed-point and are there-fore independent of time

153 PDE 7.4 Time-independent PDE-s

7.4 Time-independent PDE-s

7.4.1 Finite Difference Method (FDM)

• Discrete mesh in solving region

• Derivatives replaced with approximation by finite differences

Example. Conside Poisson equation in 2D:

−uxx−uyy = f , 0≤ x≤ 1, 0≤ y≤ 1, (19)

• boundary values as on the figure on the left:

154 PDE 7.4 Time-independent PDE-s

0

y

x

0

1

0

y

x

0

1

0

• Define discrete nodes as on the figure on right

• Inner nodes, where computations are carried out are defined with

(xi,y j) = (ih, jh), i, j = 1, ...,n

– (in our case n = 2 and h = 1/(n+1) = 1/3)

155 PDE 7.4 Time-independent PDE-s

Consider here that f = 0.Replacing 2nd order derivatives with standard 2nd order differences in mid-points,

we get

ui+1, j−2ui, j +ui−1, j

h2 +ui, j+1−2ui, j +ui, j−1

h2 = 0, i, j = 1, ...,n,

where ui, j is approximation of the real solution u = u(xi,y j) in point (xi,y j), and in-cludes one boundary value, if i or j is 0 or n+1. As a result we get:

4u1,1−u0,1−u2,1−u1,0−u1,2 = 0

4u2,1−u1,1−u3,1−u2,0−u2,2 = 0

4u1,2−u0,2−u2,2−u1,2−u1,3 = 0

4u2,2−u1,2−u3,2−u2,2−u2,3 = 0.

In matrix form:

156 PDE 7.4 Time-independent PDE-s

Ax =

4 −1 −1 0−1 4 0 −1−1 0 4 −10 −1 −1 4

u1,1

u2,1

u1,2

u2,2

=

u0,1 +u1,0

u3,1 +u2,0

u0,2 +u1,3

u3,2 +u2,3

=

0011

= b.

This positively defined system can be solved directly with Cholesky factorisation(Gauss elimination for symmetric matrix, where factorisation A = LT L is found) oriteratively. Exact solution of the problem is:

x =

u1,1

u2,1

u1,2

u2,2

=

0.1250.1250.3750.375

.

157 PDE 7.4 Time-independent PDE-s

In general case n2×n2 Laplace’s matrix has form:

A =

B −I 0 · · · 0

−I B −I . . . ...

0 −I B . . . 0... . . . . . . . . . −I0 · · · 0 −I B

, (20)

where n×n matrix B is of form:

B =

4 −1 0 · · · 0

−1 4 −1 . . . ...

0 −1 4 . . . 0... . . . . . . . . . −10 · · · 0 −1 4

.

It means that most of the elements of matrix A are zero How are such matrices called?– it is a sparse matrix

158 PDE 7.4 Time-independent PDE-s

7.4.2 Finite element Method (FEM)

Consider as an example Poisson equation

−∆u(x) = f (x), ∀x ∈Ω,

u(x) = g(x), ∀x ∈ Γ,

where Laplacian ∆ is defined by

(∆u)(x) = (∂ 2u∂x2 +

∂ 2u∂y2 )(x), x =

(xy

)

In Finite element Method the region is divided into finite elements.

159 PDE 7.4 Time-independent PDE-s

A region divided into Finite Elements:

160 PDE 7.4 Time-independent PDE-s

Consider unit square.

y

x

The problem in Variational Formulation: Find uh ∈Vh such that

a(uh,v) = ( f ,v), ∀v ∈Vh (21)

where in case of Poisson equation

a(u,v) =∫

Ω

∇u ·∇vdx

161 PDE 7.4 Time-independent PDE-s

and (u,v) =∫

Ωu(x)v(x)dx. The gradient ∇ of a scalar function f (x,y) is defined by:

∇ f =(

∂ f∂x

,∂ f∂y

)

• In FEM the equation (21) needs to be satisfied on a set of testfunctions ϕi =

ϕi(x),

– which are defined such that

ϕi =

1, x = xi

0 x = x j j 6= i

• and it is demanded that (21) is satisfied with each ϕi (i = 1, ...,N) .

• As a result, a matrix of the linear equations is obtained

162 PDE 7.5 Sparse matrix storage schemes

• The matrix is identical with the matrix from (20) !

• Benefits of FEM over finite difference schemes

– more flexible in choosing discretization

– existence of thorough mathematical constructs for proof of convergenceand error estimates

7.5 Sparse matrix storage schemes

• As we saw, different discretisation schemes give systems with similar matrixstructures

• (In addition to FDM and FEM often also some other discretisation schemes areused like Finite Volume Method (but we do not consider it here))

• In each case, the result is a system of linear equations with sparse matrix.

How to store sparse matrices?How to store sparse matrices?

163 PDE 7.5 Sparse matrix storage schemes

7.5.1 Triple storage format

• n×m matrix A each nonzero with 3 values: integers i and j and (in most appli-cations) real matrix element ai j. =⇒ three arrays:

indi(1:nz), indj(1:nz), vals(1:nz)of length nz, – number of matrix A nonzeroes

Advantages of the scheme:

• Easy to refer to a particular element

• Freedom to choose the order of the elements

Disadvantages :

• Nontrivial to find, for example, all nonzeroes of a particular row or column andtheir positions

164 PDE 7.5 Sparse matrix storage schemes

7.5.2 Column-major storage format

For each matrix A column k a vector row_ind(j) – giving row numbers i forwhich ai j 6= 0.

• To store the whole matrix, each column nonzeros

– added into a 1-dimensonal array row_ind(1:nz)

– introduce cptr(1:M) referring to column starts of each column inrow_ind.

row_ind(1:nz), cptr(1:M), vals(1:nz)

Advantages:

• Easy to find matrix column nonzeroes together with their positions

165 PDE 7.5 Sparse matrix storage schemes

Disadvantages:

• Algorithms become more difficult to read

• Difficult to find nonzeroes in a particular row

7.5.3 Row-major storage format

For each matrix A row k a vector col_ind(i) giving column numbers j forwhich ai j 6= 0.

• To store the whole matrix, each row nonzeros

– added into a 1-dimensonal array col_ind(1:nz)

– introduce rptr(1:N) referring to row starts of each row in col_ind.

col_ind(1:nz), rptr(1:N), vals(1:nz)

166 PDE 7.5 Sparse matrix storage schemes

Advantages:

• Easy to find matrix row nonzeroes together with their positions

Disadvantages:

• Algorithms become more difficult to read.

• Difficult to find nonzeroes in a particular column.

7.5.4 Combined schemes

Triple format enhanced with cols(1:nz), cptr(1:M), rows(1:nz),

rptr(1:N). Here cols and rows refer to corresponding matrix A values intriple format. E.g., to access row-major type stuctures, one has to index throughrows(1:nz)

Advantages:

167 PDE 7.5 Sparse matrix storage schemes

• All operations easy to perform

Disadvantages:

• More memory needed.

• Reference through indexing in all cases

168 Iterative methods 8.1 Problem setup

8 Iterative methods

8.1 Problem setup

Itereative methods for solving systems of linear equations withsparse matrices

Consider system of linear equations

Ax = b, (22)

where N×N matrix A

• is sparse,

– number of elements for which Ai j 6= 0 is O(N).

• Typical example: Poisson equation discretisation on n×n mesh, (N = n×n)

– in average 5, nonzeros per A row

169 Iterative methods 8.1 Problem setup

In case of direct methods, like LU-factorisation

• memory consumption (together with fill-in): O(N2) = O(n4).

• flops: 2/3 ·N3 +O(N2) = O(n6).

Banded matrix LU-decomposition

• memory consumption (together with fill-in): O(N · L) = O(n3), where L isbandwidth

• flops: 2/3 ·N ·L2 +O(N ·L) = O(n4).

170 Iterative methods 8.2 Jacobi Method

8.2 Jacobi Method• Iterative method for solving (22)

• With given initial approximation x(0), approximate solution x(k), k = 1,2,3, ...of (22) real solution x are calculated as follows:

– i-th component of x(k+1), x(k+1)i is obtained by taking from (22) only the

i-th row:Ai,1x1 + · · ·+Ai,ixi + · · ·+Ai,NxN = bi

– solving this with respect to xi, an iterative scheme is obtained:

x(k+1)i =

1Ai,i

(bi−∑

j 6=iAi, jx

(k)j

)(23)

171 Iterative methods 8.2 Jacobi Method

The calculations are in essence parallel with respect to i – no dependence onother componens x(k+1)

j , j 6= i. Iteration stop criteria can be taken, for example:∥∥∥x(k+1)−x(k)∥∥∥< ε or k+1≥ kmax, (24)

– ε – given error tolerance

– kmax – maximal number of iterations

• memory consumption (no fill-in):

– NA6=0 – number of nonzeroes of matrix A

• Number of iterations to reduce∥∥∥x(k)−x

∥∥∥2< ε

∥∥∥x(0)−x∥∥∥

2:

#IT≥ 2lnε−1

π2 (n+1)2 = O(n2)

172 Iterative methods 8.2 Jacobi Method

• flops/iteration ≈ 10 ·N = O(n2), =⇒

#IT · flopsiteration

=Cn4 +O(n3) = O(n4).

coefficent C in front of n4 is:

C ≈ 2lnε−1

π2 ·10≈ 2 · lnε−1

• Is this good or bad?This is not very good at all... We need some better methods, because

– For LU-decomposition (banded matrices) we had C = 2/3

173 Iterative methods 8.3 Conjugate Gradient Method (CG)

8.3 Conjugate Gradient Method (CG) C a l c u l a t e r(0) = b−Ax(0) wi th g i v e n s t a r t i n g v e c t o r x(0)

f o r i = 1 , 2 , . . .s o l v e Mz(i−1) = r(i−1) # we assume here t h a t M = I nowρi−1 = r(i−1)T z(i−1)

i f i ==1p(1) = z(0)

e l s eβi−1 = ρi−1/ρi−2

p(i) = z(i−1)+βi−1p(i−1)

e n d i fq(i) = Ap(i) ; αi = ρi−1/p(i)T q(i)

x(i) = x(i−1)+αip(i) ; r(i) = r(i−1)−αiq(i)

check c o n v e r g e n c e ; c o n t in u e i f neededend

174 Iterative methods 8.3 Conjugate Gradient Method (CG)

• memory consumption (no fill-in):

NA6=0 +O(N) = O(n2),

where NA6=0 – # nonzeroes ofA

• Number of iterations to achieve∥∥∥x(k)−x

∥∥∥2< ε

∥∥∥x(0)−x∥∥∥

2:

#IT≈ lnε−1

2

√κ(A) = O(n)

• Flops/iteration ≈ 24 ·N = O(n2) , =⇒

#IT · flopsiteration

=Cn3 +O(n2) = O(n3),

175 Iterative methods 8.3 Conjugate Gradient Method (CG)

where C ≈ 12lnε−1 ·√

κ2(A).

=⇒ C depends on condition number of A! This paves the way for preonditioningtechnique

176 Iterative methods 8.4 Preconditioning

8.4 PreconditioningIdea:Replace Ax = b with system M−1Ax = M−1b.Apply CG to

Bx = c, (25)

where B = M−1A and c = M−1b.But how to choose M?Preconditioner M = MT to be chosen such that

(i) Problem Mz = r being easy to solve

(ii) Matrix B being better conditioned than A, meaning that κ2(B)< κ2(A)

177 Iterative methods 8.4 Preconditioning

Then#IT(25) = O(

√κ2(B))< O(

√κ2(A)) = #IT(22)

butflops

iteration(25) =

flopsiteration

(22)+ (i) >flops

iteration(22)

• =⇒We need to make a compromise!

• (In extreme cases M = I or M = A)

• Preconditioned Conjugate Gradients (PCG) Method

– obtained if to take in previous algorithm M 6= I

178 Iterative methods 8.5 Preconditioner examples

8.5 Preconditioner examplesDiagonal Scaling (or Jacobi method)

M = diag(A)

(i) flopsIteration = N

(ii) κ2(B) = κ2(A)

⇒ Is this good?no large improvement to be expeted

179 Iterative methods 8.5 Preconditioner examples

Incomplete LU-factorisation

M = LU ,

• L and U – approximations to actual factors L and U in LU-decompoition

– nonzeroes in Li j and Ui j only where Ai j 6= 0 (i.e. fill-in is ignored in LU-factorisation algorithm)

(i) flopsIiteration = O(N)

(ii) κ2(B)< κ2(A)

How good is this preconditioner?Some improvement at least expected!

κ2(B) = O(n2)

180 Iterative methods 8.5 Preconditioner examples

Gauss-Seidel method do k=1,2,...

do i=1,...,n

x(k+1)i =

1Ai,i

(bi−

i−1

∑j=1

Ai jx(k+1)j −

n

∑j=i+1

Ai, jx(k)j

)(26)

enddo

enddoNote that in real implementation, the method is done like:

do k=1,2,...

do i=1,...,n

xi =1

Ai,i

(bi−∑

j 6=iAi jx j

)(27)

enddo

enddoDo you see a problem with this preconditioner (with PCG method)?But the preconditioner is not symmetric, which makes CG not to converge!

181 Iterative methods 8.5 Preconditioner examples

Symmetric Gauss-Seidel methodTo get the symmetric preconditioner, another step is added:

do k=1,2,...

do i=1,...,n

xi =1

Ai,i

(bi−∑ j 6=i Ai jx j

)enddo

enddo

do k=1,2,...

do i=n,...,1

xi =1

Ai,i

(bi−∑ j 6=i Ai jx j

)enddo

enddo

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