Marching Solver for Poisson Equation

大氣四 簡睦樺

Outline

• A brief review for Poisson equation and marching method

• Parallel algorithm and consideration for efficiency

• Results • Future works

Poisson Equation

• A second order partial differential equation with inhomogeneous forcing term is called Poisson equation. i.e.

• Ω is the domain, L is some operator for boundary condition.

2

2 for , 1,2

0 for

i

i

u f u ix

L u u

Numerical approximation

• Let Uij=u(x,y), approximate the equation with finite difference method.

• For marching method

1, , 1, , 1 , , 1,2 2

2 2i j i j i j i j i j i ji j

U U U U U Uf

x y

2

2, 1 , , 1, , 1, , 122 2i j i j i j i j i j i j i j

yU y f U U U U Ux

Marching method

• Let , where Ui,j is the exact solution, U’i,j is guess solution, ei,j is the difference between Ui,j and U’i,j.

• ij=0…n+1(Ui,0 , U0,j ,Ui,n+1 and Un+1,j are boundary)

• We have a initial guess for U’i,1 and compute U’i,2 via marching equation.

, , ,i j i j i jU U e

• Let a vector of equation of e, i.e. E1,1 =[1 0 ...0]T and E2,1 =[0 1 0 ...0]T and compute the equation of e.

• Thus, we can construct and system of ein+1

and U’in+1 –Uin+1. i.e.

• We solve the system and get ei,1 , so we can get Ui,1

2

, 1 , 1, , 1, , 122 2i j i j i j i j i j i jyE E E E E Ex

, 1 , 1 , 1 ,1 1, 1 1, 1 ,1...i n i n i n i n n n iU U e Ce E E e

Domain decomposition idea

• However, the above method may fail easily because the condition number of the system may be to huge.

• To construct error vector from Ei,1 , En+1,1 and other interior point.

• we can have the local equation for some marching points for which different initial condition march and should have same value.

Parallel Algorithm

• Use MPI to compute the decomposed domain.

• Separate the computation into preprocessor and marching.

Preprocessor

• To compute inverse matrix of the systempreprocessor

Compute error vector

Compute error vector

Parallel LU Parallel LU

Out put Out put

Marching

marching

Read data Read data

marching marching

Collect errorvector

Matrix product Matrix product

ConnectBoundary data

Marching again to get exact sol.

Marching again to get exact sol.

Operation count

• For general case, define nx for grid points at x-direction, ny for grid points at y direction, nb is number of sub-domain ～ nx/5.

• For serial case, preprocessor:Error vector: O(ny*ny*nx) LU: O((ny*nb)3)

marching:marching: O(ny*nx)

matrix product: O((ny*nb)2)

Parallel case

• ng is number of MPI world. nb>=ng• preprocessor:

Error vector: O(ny*ny*nx/ng) LU: O((ny*nb)3)

marching:marching: O(ny*nx/ng)

matrix product: O((ny*nb/ng)2)

Compare with other method

Serial March LU Iteration+prep

Iteration

Prep (ny*nb)3 (ny*nx)3 (ny*nx)3 0

march (ny*nb)2 (ny*nx)2 (ny*nx)2 (ny*nx)3

1 time problem

(ny*nb)3 (ny*nx)3 (ny*nx)3 (ny*nx)3

results

• Converge error• Efficiency improvement• Annulus testing case

Converge error

2 2

2 2 1.25exp 2

u u x yx yDirichlet BC

• Testing case

Error table

H Error Ratio order

0.0625 4.987e-5 NaN NaN

0.03125 1.328e-5 3.75527 1.90892

0.1563 3.425e-6 3.87737 1.95508

0.00781 8.697e-7 3.93814 1.97751

Log-log plot

Efficiency• For 128*128 case NP Time

1 0.0412

2 0.0345

4 0.0181

8 0.0134

16 0.0123

Annulus

Future work

• Parallel LU• 3D marching method