Presentation on theme: "Computational Methods II (Elliptic)"— Presentation transcript:
1Computational Methods II (Elliptic) Week 4- Lecture 1 and 2Computational Methods II (Elliptic)Dr. Farzad IsmailSchool of Aerospace and Mechanical EngineeringUniversiti Sains MalaysiaNibong Tebal Pulau Pinang
2Overview We already know the nature of hyperbolic and parabolic PDE’s. Now we will focus on elliptic PDE.The elliptic PDE is a type of PDE for solving incompressible flow.
3Overview (cont’d) Elliptic PDE’s are always smooth Easier to obtain accurate solutions compared to hyperbolic problems, the challenge is in getting solutions efficiently.Unlike hyperbolic and parabolic problems, information is transmitted everywhere
4Overview (cont’d) The model problem that will be discussed is Note that the problem is now 2D but with no time dependencePoisson EqnLaplace Equation
6Overview (cont’d)For special case where the grid sizes in x and y is identicalSolve for u(j,k) such that r(j,k)=0, yielding a system ofmatrix Mu = 0Assume M intervals for each direction, Gaussian elimination method needs operations in 2Dis the residual associated with cell (j,k)
7Iteration MethodSolving it directly is to expensive, usually an iteration method is employed.The simplest iteration method is the point Jacobi method – based on solving the steady state of parabolic pdeSolve iteratively for until
8Iteration Method (cont’d) The superscript n denotes pseudo-time, not physical timeω is the relaxation factor, analogous to μThe first thought is to achieve steady-state as quickly as possible, hence choosing a largest ω consistent with stabilityNot a good idea since we need to know what is the best choice for ω
10Residual Pattern Plot of convergence history Has three distinct phases First phase, the residual decays very rapidlySecond phase decays linearly on the log-plot (or exponentially in real plot)Third phase is ‘noise’, since randomness has set in- after a while residuals become so small that theyare of the order of round-off errors
11Residual Pattern (cont’d) Sometimes the residual plot would ‘hang’ upUsually due to a ‘bug’ in the codeTo save time, apply a ‘stopping’ criterion to residualBut not easy to do so, need to understand errors
12Analyzing the Errors We want know how the iteration errors decay. Expand the solution asThe first term on the RHS is the solution after infinite number of iterationsThe second term on the RHS is the error between the solution at iterative level n and after infinite iterationsGet the best solution if error is removed
13Analyzing the Errors (cont’d) Substitute the error relation into the iteration method.Converge solution gives zero residual, henceIterations for error
14Analyzing the Errors (cont’d) Shows that error itself follow an evolutionary patternThis is true for all elliptic problemsRemarkably, we can determine how the error would behave even if we do not know the solution
15Analyzing the Errors using VA In 2D, von Neumann analysis (VA)Insert this into the error equationYields the amplification factor for the errors
16Amplification factor for Point Jacobi Method What does the figure tell you ???
17G for Errors using Point Jacobi (cont’d) The plot shows how the errors decay for various error modes (in terms of sines and cosines)Consider 1D situation – let there be M+1 points on a line giving M intervals. The errors can be expressed aswhere m = 1,2, .., M-1
18Error Modes for M=8 Seven possible error modes on a 1D mesh with eight intervals –Lowest to highest frequencies.
19Error Modes Analysis M=8 Any numbers satisfying f(0)=f(M)=0 can be represented asThis is a discrete Fourier TransformIn 1D, need only to consider the following set of discrete frequencies (lowest to highest)
202D situationConsider 2D situation – let there be M+1 and N+1 points, giving M and N intervals in x and y directionswhere m = 1,2, .., M-1. n=1,2,.. N-1.Assume M=N, the highest and lowest frequencies correspond to
21Discrete G versus z - Each point corresponds to discrete error modes, each error mode decays differently even for samerelaxation factor- Note ZL = -ZR
22Decay of Error ModesIn eliminating the errors, we do not care the sign of amplitude, just the magnitude of it – want g as small as possibleIf 0 < |g| < 1 for ALL error modes, then we could say that we eventually will remove all the iteration errors and hence a steady state solution will be obtained.However, the errors will be removed at different rates, with modes corresponding to the largest |g| (convergence rate) will be most difficult to be removed- dominant error mode
23Decay of Error Modes (cont’d) For ω=0.6, it is clear that ZR (lowest frequency) is the dominant error modeIncreasing ω, will decrease g until ω=1 – both lowest and highest frequency modes become dominantBeyond ω > 1, ZL (highest frequency) modes become more dominant and |g| deteriorates.
24Computational CostRecall for direct method (Gaussian Elimination) requires in 2D and in 3D.Using point Jacobi iteration methods costsin 2D and in 3D.Can this be improved? Let say by one order magnitude?
25Gauss Seidel When visiting node (j,k) at time level n+1, can use information of nodes (j,k-1) at time level n and(j-1, k) at n+1 (square nodes)- This is Gauss-Seidel method
26SOR MethodCoupling Gauss-Seidel with point Jacobi gives Successive Over- Relaxation (SOR) methodIt can be shown that using VA, the convergence rateOnly consider real z such that
27G versus z for SOR - Note now z = [0,2] Similar to Point Jacobi for ω less than or equal to 1But improved convergence rate for ω > 1 (1.16, 1.36, 1.64,1.81 ) -> STILL LOWEST FREQUENCY MODE IS DOMINANT
28Discussion of Low Frequency Modes We have seen that the low frequency modes are most difficult to get rid of for Point Jacobi and SOR methodsModerate to high frequency modes are usually much easier to be removed.This is the general behavior for almost all methods in solving elliptic PDENeed to think out of the box
29Multigrid (Brandt)Use a hierachy of different grid sizes to solve the elliptic PDE –splitting the error modesThe low frequency error modes on a fine grid will appear as high frequency modes on a coarser grid.Hence easier to be removedBut there is a problem with coarse grids.
30Multigrid MeshAn 8 x 8 grid superimposed with 4 x x 2 grid
31Multigrid (cont’d)Although coarse grids will remove the error modes quickly, but it will converge to an inaccurate solutionNeed to combine the accuracy of fine grids with the fast convergence on coarse gridsAs long as there are significant error modes in fine grid, coarse grid must be kept to workFine grids must ‘tell’ the coarse grids the dominant error modes and ask ‘advice’ on how to remove them – restriction operator
32Multigrid (cont’d)Coarse grids will give information on how to remove the dominant error modes to fine grids and ask for ‘advice’ regarding accuracy- prolongation operatorCommunication between coarse-fine grids is the keyIn between, perform relaxation (solving the steady state equation)