An Iterative Solver for the Diffusion The Methods Progress So - PowerPoint PPT Presentation

An Iterative Solver for the Diffusion Equation Alan Davidson Problem Statement An Iterative Solver for the Diffusion The Methods Progress So Far... Equation Alan Davidson adavidso@cs.hmc.edu 25 April 2006

An Iterative Solver The Diffusion/Heat Equation for the Diffusion Equation Alan Davidson Problem Statement The Methods Progress So Far... u t = a + D · u xx • u is the concentration/temperature • a is a source/sink • D is a diffusion/thermal diffusivity constant • t is time, x is space

An Iterative Solver The Hitch for the Diffusion Equation Alan Davidson Problem Statement The Methods Progress So Far... • For clinic, we needed arbitrary Dirichlet boundary conditions through the middle • These BCs simply hold the concentration at a fixed amount • Exact solution cannot be found easily

An Iterative Solver The Solution! for the Diffusion Equation Alan Davidson Problem Statement • Use an iterative solver The Methods • For clinic we actually used Progress So Far... • The 3-D equation • The Gauss-Seidel method • The backwards Euler FDA • C++ • I wanted to try • 1, 2, or 3 dimensions • Dirichlet, Neumann, and Cauchy BCs • The Jacobi or SSOR methods • The backwards Euler FDA • Sparse matrices in Matlab

An Iterative Solver Iterative Methods for the Diffusion Equation Alan Davidson Problem We want to solve Statement The Methods � = A � x b Progress So Far... but we don’t want to invert A (time constraints, etc). Divide up A so that = D + L + U A where • D is diagonal • L is lower triangular • U is upper triangular

An Iterative Solver The Jacobi Method for the Diffusion Equation Alan Davidson Problem Statement The Methods Progress So Far... x (0) to be the initial guess at a solution. Now, define Pick � x ( i ) + D − 1 · � x ( i +1) D − 1 · ( − L − U ) · � � = b � isn’t small enough, repeat. � x ( i +1) − � x ( i ) � If � �

An Iterative Solver The Successive Over-Relaxation (SOR) Method for the Diffusion Equation Alan Davidson We will solve Problem Statement ω� ω A � = x b The Methods Progress So Far... Noting that = ( D + ω L ) + ( ω U − (1 − ω ) D ) , ω A we now have that � x ( i ) + ω� � x ( i +1) ( D + ω L ) − 1 · � = ( − ω U + (1 − ω ) D ) � b The backwards version is � x ( i ) + ω� � x ( i +1) ( D + ω U ) − 1 · = ( − ω L + (1 − ω ) D ) � � b

An Iterative Solver The Symmetric Successive Over-Relaxation for the Diffusion Equation (SSOR) Method Alan Davidson Problem Statement The Methods Progress So Far... We do one forwards SOR step followed by one backwards SOR step: � x ( i ) + ω� � x ( i +1 / 2) ( D + ω L ) − 1 · � = ( − ω U + (1 − ω ) D ) � b � x ( i +1 / 2) + ω� � x ( i +1) ( D + ω U ) − 1 · � = ( − ω L + (1 − ω ) D ) � b

An Iterative Solver Progress So Far... for the Diffusion Equation Alan Davidson Problem Statement Slower than I expected The Methods Progress So Far... • 1-D case implemented for both Jacobi and SSOR methods with any Dirichlet BCs • Neumann (and therefore Cauchy) BCs are not well defined in arbitrary locations, especially in a 1-D case • Uses only sparse matrices • Checks for stability Oddly enough, the Jacobi method seems to converge more quickly than the SSOR!?

An Iterative Solver An Example Run... for the Diffusion Equation Alan Davidson Problem Statement The Methods Progress So Far...

An Iterative Solver An Example Run... for the Diffusion Equation Alan Davidson Problem Statement The Methods Progress So Far...

An Iterative Solver An Example Run... for the Diffusion Equation Alan Davidson Problem Statement The Methods Progress So Far...

An Iterative Solver An Example Run... for the Diffusion Equation Alan Davidson Problem Statement The Methods Progress So Far...

An Iterative Solver An Example Run... for the Diffusion Equation Alan Davidson Problem Statement The Methods Progress So Far...

An Iterative Solver An Example Run... for the Diffusion Equation Alan Davidson Problem Statement The Methods Progress So Far...

An Iterative Solver An Example Run... for the Diffusion Equation Alan Davidson Problem Statement The Methods Progress So Far...

An Iterative Solver References Used for the Diffusion Equation Alan Davidson Problem Statement The Methods Progress So Far... Saad, Yousef. Iterative Methods for Sparse Linear Systems SIAM, 2000. http://mathworld.wolfram.com

An Iterative Solver That’s All, Folks! for the Diffusion Equation Alan Davidson Problem Statement The Methods Progress So Far... Questions?