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

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*Donald J. Estep; Mats G. Larson; Roy D. Williams*

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.

#### Table of Contents

# Table of Contents

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

- Contents vii8 free
- Chapter 1. Introduction 110 free
- Chapter 2. A framework for a posteriori error estimation 1726
- Chapter 3. The size of the residual errors and stability factors 3039
- Chapter 4. Computational error estimation 4251
- 4.1. Two examples and a stability factor gallery 4251
- 4.2. Choosing data for the dual problem 4554
- 4.3. Linearization and the approximate dual problem 5059
- 4.4. A test of the accuracy and reliability of the error estimate 5362
- 4.5. Some details of implementation 5463
- 4.6. Numerical results for the nine models 6069

- Chapter 5. Preservation of invariant rectangles under discretization 7483
- Chapter 6. Details of the analysis in Chapter 2 8796
- Chapter 7. Details of the analysis in Chapter 3 91100
- Chapter 8. Details of the analysis in Chapter 5 99108
- Bibliography 106115