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Estimating the Error of Numerical Solutions of Systems of Reaction-Diffusion Equations

Donald J. Estep Georgia Institute of Technology, Atlanta, GA
Mats G. Larson Chalmes University of Technology, Goteborg, Sweden
Roy D. Williams California Institute of Technology, Pasadena, CA
Available Formats:
Electronic ISBN: 978-1-4704-0287-7
Product Code: MEMO/146/696.E
109 pp
List Price: $52.00 MAA Member Price:$46.80
AMS Member Price: $31.20 Click above image for expanded view Estimating the Error of Numerical Solutions of Systems of Reaction-Diffusion Equations Donald J. Estep Georgia Institute of Technology, Atlanta, GA Mats G. Larson Chalmes University of Technology, Goteborg, Sweden Roy D. Williams California Institute of Technology, Pasadena, CA Available Formats:  Electronic ISBN: 978-1-4704-0287-7 Product Code: MEMO/146/696.E 109 pp  List Price:$52.00 MAA Member Price: $46.80 AMS Member Price:$31.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 1462000
MSC: Primary 65; 35;

This paper is concerned with the computational estimation of the error of numerical solutions of potentially degenerate reaction-diffusion equations. The underlying motivation is a desire to compute accurate estimates as opposed to deriving inaccurate analytic upper bounds. In this paper, we outline, analyze, and test an approach to obtain computational error estimates based on the introduction of the residual error of the numerical solution and in which the effects of the accumulation of errors are estimated computationally.

We begin by deriving an a posteriori relationship between the error of a numerical solution and its residual error using a variational argument. This leads to the introduction of stability factors, which measure the sensitivity of solutions to various kinds of perturbations. Next, we perform some general analysis on the residual errors and stability factors to determine when they are defined and to bound their size. Then we describe the practical use of the theory to estimate the errors of numerical solutions computationally. Several key issues arise in the implementation that remain unresolved and we present partial results and numerical experiments about these points. We use this approach to estimate the error of numerical solutions of nine standard reaction-diffusion models and make a systematic comparison of the time scale over which accurate numerical solutions can be computed for these problems. We also perform a numerical test of the accuracy and reliability of the computational error estimate using the bistable equation. Finally, we apply the general theory to the class of problems that admit invariant regions for the solutions, which includes seven of the main examples. Under this additional stability assumption, we obtain a convergence result in the form of an upper bound on the error from the a posteriori error estimate. We conclude by discussing the preservation of invariant regions under discretization.

Researchers interested in solving nonlinear diffusion equations and in error analysis of numerical methods for differential equations.

• Chapters
• 1. Introduction
• 2. A framework for a posteriori error estimation
• 3. The size of the residual errors and stability factors
• 4. Computational error estimation
• 5. Preservation of invariant rectangles under discretization
• 6. Details of the analysis in Chapter 2
• 7. Details of the analysis in Chapter 3
• 8. Details of the analysis in Chapter 5
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Volume: 1462000
MSC: Primary 65; 35;

This paper is concerned with the computational estimation of the error of numerical solutions of potentially degenerate reaction-diffusion equations. The underlying motivation is a desire to compute accurate estimates as opposed to deriving inaccurate analytic upper bounds. In this paper, we outline, analyze, and test an approach to obtain computational error estimates based on the introduction of the residual error of the numerical solution and in which the effects of the accumulation of errors are estimated computationally.

We begin by deriving an a posteriori relationship between the error of a numerical solution and its residual error using a variational argument. This leads to the introduction of stability factors, which measure the sensitivity of solutions to various kinds of perturbations. Next, we perform some general analysis on the residual errors and stability factors to determine when they are defined and to bound their size. Then we describe the practical use of the theory to estimate the errors of numerical solutions computationally. Several key issues arise in the implementation that remain unresolved and we present partial results and numerical experiments about these points. We use this approach to estimate the error of numerical solutions of nine standard reaction-diffusion models and make a systematic comparison of the time scale over which accurate numerical solutions can be computed for these problems. We also perform a numerical test of the accuracy and reliability of the computational error estimate using the bistable equation. Finally, we apply the general theory to the class of problems that admit invariant regions for the solutions, which includes seven of the main examples. Under this additional stability assumption, we obtain a convergence result in the form of an upper bound on the error from the a posteriori error estimate. We conclude by discussing the preservation of invariant regions under discretization.