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A posteriori error estimates for the exponential midpoint method for linear and semilinear parabolic equations

Xianfa Hu, Wansheng Wang, Mengli Mao, Jiliang Cao

Abstract

In this paper, the a posteriori error estimates of the exponential midpoint method for time discretization are studied for linear and semilinear parabolic equations. Using the exponential midpoint approximation defined by a continuous and piecewise linear interpolation of nodal values yields the suboptimal order estimates. Based on the property of the entire function, we introduce a continuous and piecewise quadratic time reconstruction of the exponential midpoint method to derive the optimal order estimates, and the error bounds are solely dependent on the discretization parameters, the data of the problem and the approximation of the entire function. Several numerical examples are implemented to illustrate the theoretical results.

A posteriori error estimates for the exponential midpoint method for linear and semilinear parabolic equations

Abstract

In this paper, the a posteriori error estimates of the exponential midpoint method for time discretization are studied for linear and semilinear parabolic equations. Using the exponential midpoint approximation defined by a continuous and piecewise linear interpolation of nodal values yields the suboptimal order estimates. Based on the property of the entire function, we introduce a continuous and piecewise quadratic time reconstruction of the exponential midpoint method to derive the optimal order estimates, and the error bounds are solely dependent on the discretization parameters, the data of the problem and the approximation of the entire function. Several numerical examples are implemented to illustrate the theoretical results.
Paper Structure (21 sections, 2 theorems, 120 equations, 11 tables)

This paper contains 21 sections, 2 theorems, 120 equations, 11 tables.

Table of Contents

  1. Introduction
  2. The exponential midpoint method for parabolic problems
  3. Assumptions and notations
  4. Exponential Runge-Kutta methods of collocation type
  5. The exponential midpoint method for parabolic problems
  6. A posteriori error estimates for linear problems
  7. Exponential midpoint rule
  8. Maximal norm estimate
  9. ${L^2(0,t;V)}$-estimate
  10. Exponential midpoint reconstruction
  11. Optimal order a posteriori error estimates for linear problems
  12. ${L^2(0,T;V)}$-estimate
  13. ${L^{\infty}(0,t;H)}$-estimate
  14. Estimate of ${\hat{U}-U}$
  15. Error estimates for semilinear parabolic differential equation
  16. ...and 6 more sections

Key Result

Theorem 1

Let $u(t)$ be the exact solution of linear problem, and $U(t)$ is the exponential midpoint approximation to $u(t)$ defined by linear interpolation. Denoting $R(t)$ as the residual of $U(t)$, then the error $e(t)=u(t)-U(t)$ satisfies

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Example 1
  • Example 2
  • Example 3
  • Example 4