MATH 447 (Numerical Analysis II)

MATH 447 (Numerical Analysis II)

Project 4: Burgers' Equation (Nonlinear Partial Differential Equations)

===> Ahmad Amin, Brandon Scott, Mike Sullivan <===

To numerically solve Burgers' Equation (Nonlinear PDE) with Dirichlet Boundary Conditions requires discretization via the following techniques:
  1. Backward-Difference Method
  2. Center-Difference Method
  3. Multivariate Newton Method
    • Jacobian
    • Backslash (Solve "Systems of Equations") $$Ax=b$$

Detailed Methodology: Numerical Methods for Solving Burgers Equation (Nonlinear PDE)


BURGERS' EQUATION (FLUID FLOW MODEL) WITH DIRICHLET BOUNDARY CONDITIONS
Task # =====================>Description<===================== Solution Matlab Code Plot (.pdf) Plot (.png) ========>Notes<=========
1 Exercise 8.4.3: Eqn 8.68: Exact Solution to Burgers' Eqn Exact Solution Proof --- --- --- ---
2 Example 8.12 (P440): Baseline --- burgers.m burgers_00.pdf burgers_00.png w = burgers(0,1,0,2,20,40) h=0.05; k=0.05; ===> [21 x 21 Matrix]
3 Computer Problem 8.4.1(a) --- burgers_01a.m burgers_01a.pdf burgers_01a.png w=burgers_01a(0,1,0,1,10,10) h=0.1; k=0.1 ===> [11 x 11 Matrix]
4 Computer Problem 8.4.1(b) --- burgers_01b.m burgers_01b.pdf burgers_01b.png w=burgers_01b(0,1,0,1,50,50) h=0.02; k=0.02 ===> [51 x 51 Matrix]
5 Computer Problem 8.4.2(a) --- burgers_02a.m burgers_02a1.pdf burgers_02a1.png w=burgers_02a(0,1,0,2,100,32) h=0.01; k=0.0625 ===> [101 x 16 Matrix]
6 Computer Problem 8.4.2(b) --- errortimestep_code.m errortimestep.pdf errortimestep.png ===> SEE TABLE BELOW







APPROXIMATION ERRORS WITH DIRICHLET BOUNDARY CONDITIONS
X T P K (WHERE K = 2-p) N Console Code Computed Value Exact Value Approximation Error
1/2 1 4 0.0625 32 burgers(0,1,0,2,100,32) 0.168232601865159 0.130920658614977 0.037311943250182
1/2 1 5 0.03125 64 burgers(0,1,0,2,100,64) 0.149024547245264 0.130920658614977 0.018103888630288
1/2 1 6 0.015625 128 burgers(0,1,0,2,100,128) 0.139753707450387 0.130920658614977 0.008833048835411
1/2 1 7 0.0078125 256 burgers(0,1,0,2,100,256) 0.135204659805387 0.130920658614977 0.004284001190411
1/2 1 8 0.00390625 512 burgers(0,1,0,2,100,512) 0.132952086182673 0.130920658614977 0.002031427567696