Verification Techniques for BVP

Note: There are fewer commercial codes for BVP than IVP which put additionalresponsibility on the scientist to ensure the validity of numerical solutions.

Numerical Methods for BVP

Recall:

Grid:

Standard Numerical Techniques:

• Finite difference

• Galerkin (finite element)

• Shooting

Verification Strategies:

• Method of manufactured solutions

• Compare multiple methods

• Compare multiple stepsizes

• Method of nearby problems

Verification Techniques for BVP -- Manufactured Solutions

Example: Suppose we want codes for problems of the form

Notes:• This provides analytic solution to test convergence rates for numerical methods

• If possible, avoid coefficients of 1

• Smoothness of chosen solution should be commensurate with that of finalproblem

Verification Techniques for BVP -- Multiple Methods

Problem: Suppose we do not have an analytic solution

Example:

Strategies:

• Compare multiple methods

• Compare multiple stepsizes

• Method of nearby problems

Verification Techniques for BVP -- Multiple Methods

Example:

Finite Difference: See Lecture 3

System:

Matrix System:

Verification Techniques for BVP -- Multiple Methods

Example:

Galerkin: See Lecture 3

Matrix System:

Approximate Solution:

Integrals: Gaussian quadrature; e.g., 2 pt

Issues: Must maintain accuracy whenapproximating integrals

Verification Techniques for BVP -- Checks

Check:

•Symmetry

•Boundary conditions

•Qualitative behavior

Previous Example: Is this finite difference solution correct?

BVP Verification -- Finite Difference with Two Stepsizes

Previous Example: N = 10 and N = 100

BVP Verification -- Finite Difference versus Finite Element

Previous Example: Finite difference and finite element with N=10

BVP Verification -- Nearby Problem

Previous Problem: Fit with the polynomial

BVP Verification -- Nearby ProblemPrevious Example:

Pseudo-Problem: 2nd-order polynomial

where

Two Systems:

Note: Use alternative method (e.g., symbolic)when comparing RHS

2nd-Order Polynomial: N = 10

BVP Verification -- Nearby ProblemPrevious Example:

8th-Order Polynomial:

8th-Order Polynomial: N = 10

Notes:

• Polynomialobtained withpolyfit.m may notsatisfy BC

• It is easy todifferentiate resultsfrom polyfit.m

• Be careful of highorder polynomials ---piecewise is safer!!

Errors: 8th-order polynomials

Spline versus Polynomial FitsExample: Consider the HIV model from Adams et al., 2005

Spline Fit 10th Order Polynomial Fit