16. Solution of elliptic partial differential equation

Transcription

1 16. Solution of elliptic partial differential equation

2 Recall in the first lecture of this course.

3 Assume you know how to use a computer to compute; but have not done any serious numerical computations and only know some very basic numerics. This course will teach you FORTRAN programing language to compute good habits to write structural Fortran programs to write test driver to test the Fortran program written by you or others to use well developed and optimized libraries such as LAPACK FFTW to do parallel computing in multi-process computer using OpenMP to use the these skills to solve partial differential equations governing simple physical systems such as Poisson equation wave equation heat equation.

4 and using these basic blocks plus some extra hard work eventually you are able to develop your own numerical program to simulate the more realistic physical system such as

5 Numerical simulation of flow motion A popular tool due to the burst in power of computing. Substitute for experiment when measurement is inaccessible.

6 Governing equations for motion of an incompressible fluid Property variables to describe the flow: velocity: + + pressure: ( ) Equations governing the flow: incompressibility of fluid (solenoidal condition): momentum conservation: ( ) unknowns with 4 equations

7 Numerical solution based on explicit scheme Harlow F.H. & Welch J.E Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8(12) Define velocity and pressure at discrete time instance Δ : and 0 0 ( ) Δ ( ) i.e. after obtaining and at marching towards to get But 1. How is the solenoidal condition 0 satisfied? 2. What is the equation for the pressure? Take divergence of the discretized momentum equation: Δ ( )

8 Take divergence of the time-discretized momentum equation: Δ Δ ( ) Ideally 0 and 0 since the solenoidal condition 0 must be satisfied at every time instance. But due to numerical approximation the solenoidal condition may not be satisfied exactly. Let say there is a small value of at i.e. 0 but 0. We then project that 0 at. Δ 1 Δ ( ) This is an elliptic-type partial differential equation called Poisson equation: + + So the solenoidal condition at is satisfied by solving the pressure Poisson equation. This means that pressure is a tuning property to make sure the flow remain solenoidal.

9 Discrete Fourier Transform in Higher Dimension One-dimensional: Two-dimensional:

10 Numerical Solution of Poisson Equation Elliptic-type partial differential equation for in a rectangular domain: + Discrete representation of the continuous function: ( ) ( ) Numerical solution of the equation means finding discrete satisfies the discretized equation:

11 Rectangular domain with periodic conditions in both directions periodic periodic + periodic periodic Discrete can be represented by a discrete Fourier series with the coefficients to be determined: Similarly Since are given are known. 2 ( )

12 2 2 Substitute and into Poisson equation: Only need to consider 0~ /2 and 0~ /2 for real DFT. This is valid when 0 and 0. 0 and 0 corresponds to the constant mode. The solution of the Poisson equation subject to periodic boundary conditions in both direction is indeterminant i.e. any constant can be the solution. For the well posedness of the problem 0.

13 Rectangular domain with periodic condition in one direction and Dirichlet conditions on the other boundaries ( ) given periodic + periodic 0 ( ) given Unknown: + Boundary conditions: 0 + given ( ) + given

14 Since is periodic in direction only it can be represented as: 2 ( ) ( ) periodic + periodic Similarly

15 Substitute the expansions of and into the Poisson equation: Substitute the expansions of and into the boundary conditions: This is an ordinary differential equation for subject to the upper and lower boundary conditions of Dirichlet type for different. Only need to consider 0 to /2 for real DFT. 0 0

16 Discretize at Δ Δ / 0~. The unknowns are 0~ /2 1~ If use 2nd-order finite-difference scheme to approximate the differentiation of : ( ) 2 + Δ + Δ Δ + Δ (0) Δ Δ

17 Discretized equation for : Δ + Δ Δ + Δ Δ Δ The discretized equation can be represented as a system of linear equation: Δ Δ

18 Δ Δ The ( ) matrix is tridiagonal. The linear system can be solved efficiently for various. Only need to consider 0 to /2 for real DFT. In the above system of equation is real but and are complex. So the real and imaginary parts of the unknown vector can be solved separately as: and

19 Numerical implementation Given the source function of the Poisson equation and the lower and upper boundary values and where 0 ~ ( ) and 1 ~ ( ) Call FFT to compute and Loops of for calling FFT to compute Loops of 0 ~ /2 for solving for Loops of 1 ~ to construct the tridiagonal matrix for each end loops in Call suitable solver to solve and for of each end loops of Loops of for calling inverse FFT to get

20 Rectangular domain with periodic condition in one direction and Neumann conditions on the other boundaries ( ) given periodic + periodic 0 ( ) given Unknown: + Boundary conditions: 0 + given + given

21 Since is periodic in direction only it can be represented as: 2 ( ) ( ) periodic + periodic Similarly

22 Substitute the expansions of and into the Poisson equation: Substitute the expansions of and into the boundary conditions: 0 Again this is an ordinary differential equation for subject to the upper and lower boundary conditions of Neumann type for different. Only need to consider 0 to /2 for real DFT. 0

23 Discretize at Δ Δ / 0~ the unknowns are 0 to /2. If use 2nd-order finite-difference scheme to approximate the differentiation of : + 1 2Δ + 2Δ 2 + Δ + 2 Δ 2Δ Δ + Δ 1 0 2Δ 2Δ Δ Δ + 2Δ

24 Discretized equation for : Δ + 2 Δ 2Δ Δ + Δ Δ Δ + 2Δ The discretized equation can be represented as a system of linear equation: Δ + 2Δ Δ 2Δ The ( + 1) + 1 matrix is tridiagonal. The linear system can be solved efficiently for various. Only need to consider 0 to /2 for real DFT.

25 Dirichlet conditions on upper and lower boundaries: Δ Δ Neumann conditions on upper and lower boundaries: Δ + 2Δ Δ 2Δ

26 Δ + 2Δ Δ 2Δ For 0 i.e. the constant Fourier mode the coefficient matrix of the above system of equation is singular i.e. det 0. For example ( ) given To fix the problem i.e. to make the solution unique a given constant is specified at the grid point. For example (00) 0. periodic + periodic (00) 0 ( ) given

27 1 1 R 0 To implement the solvability condition 00 0: For 0 2 solutions of 0 modes are treated after solving for 0.

28 After obtain the other can be evaluated: + 1 2Δ + 2Δ + Δ 2 Δ Δ 2 + Δ 2 + Δ 2Δ 2Δ Δ Δ 2

29 Numerical implementation Given the source function of the Poisson equation and the lower and upper boundary derivative values and where 0 ~ and 0 ~ Call FFT to compute Loops of for calling FFT to compute Loops of 1 ~ /2 for solving for Loops of 0 ~ to construct the tridiagonal matrix for each end loops in Call suitable solver to solve and for of each end loops of The constant modes need to be treated after solving 0 Compute Loops of 1 ~ to compute for 0 end loops in Loops of for calling inverse FFT to get

30 Two approaches to test the Poisson solver The first approach: Invent a known analytical function and compute analytically the right-hand-side source function of the Poisson equation ( ): + and the boundary conditions: 0 0 or Given the analytical source function and the boundary conditions at and the Poisson equation is then solved for and the numerical solution is compared with the known function. + Such an approach can be used to test the convergence of the solver by increase the grid points.

31 The second approach: Generate the solutions using random numbers. The source function and the boundary conditions are then computed numerically using spectral scheme in the periodic direction and finite-difference scheme in the non-periodic direction. Given the source function numerically: spectral + finite-difference Also generate the boundary conditions using random numbers: and the boundary conditions the Poisson equation is then solved The numerical results should be equal to the original given random numbers since the forward (computing the source function) and inverse (solving the equation) operators are identical. Such an approach is for debugging the code. + or finite-difference

32 Homework Write a subroutine solving Poisson equation in a rectangular domain as shown in the figure below with periodic boundary conditions in the direction Dirichlet condition on the upper boundary and Neumann condition on the lower boundary. Use routine in Lapack to solve the tridiagonal system of linear equation (eg dgtsv). Use FFFTW to do discrete Fourier transform. In calling the subroutine the following data are input : the lengths the rectangular domain in and directions: and the numbers of discrete grids in and directions: and the right-hand side of the Poisson equation: 0 ~ ( ) 0 ~ the values on the lower and upper boundary conditions: and 0 ~ ( ) Write a test driver (use the second approach) to assess the maximum error calling the subroutine to confirm if the Poisson solver has been implemented correctly. Use random numbers between and 1. ( ) periodic in ( ) periodic in

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