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An adaptive integrating factor midpoint method for second order evolution equations

Xianfa Hu, Fazhan Geng, Wansheng Wang

Abstract

In this paper, we consider the integrating factor midpoint method for wave-type equations and derive optimal order a posteriori error estimates. We first introduce an integrating factor midpoint approximation defined by the piecewise linear approximate solutions, and derive suboptimal order residual-based error estimates using the energy technique. Hence the key is introducing a continuous, piecewise quadratic time reconstruction to establish optimal order error bounds. Based on the reliable a posteriori error control, we develop an adaptive time-stepping strategy. Numerical examples are implemented to verify the convergence rate of an error estimator and the high efficiency of the adaptive algorithm.

An adaptive integrating factor midpoint method for second order evolution equations

Abstract

In this paper, we consider the integrating factor midpoint method for wave-type equations and derive optimal order a posteriori error estimates. We first introduce an integrating factor midpoint approximation defined by the piecewise linear approximate solutions, and derive suboptimal order residual-based error estimates using the energy technique. Hence the key is introducing a continuous, piecewise quadratic time reconstruction to establish optimal order error bounds. Based on the reliable a posteriori error control, we develop an adaptive time-stepping strategy. Numerical examples are implemented to verify the convergence rate of an error estimator and the high efficiency of the adaptive algorithm.
Paper Structure (10 sections, 2 theorems, 71 equations, 2 figures, 6 tables, 1 algorithm)

This paper contains 10 sections, 2 theorems, 71 equations, 2 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. IF midpoint method for linear problems
  3. IF midpoint approximation
  4. Suboptimal a posteriori error analysis
  5. Optimal a posteriori error estimate
  6. Reconstruction
  7. Error analysis
  8. An adaptive algorithm
  9. Numerical experiments
  10. Conclusions

Key Result

Theorem 2.1

Let $u$ be the exact solution of linearhyperbolicequation, $z=(u,v)^{\intercal}$ satisfies linearproblem, and $Z=(U,V)^{\intercal}$ be the approximation to $z$ defined by linearinterpolation. Denoting the error by $e:=z-Z=(e_U,e_V)^{\intercal}$, for any $0<\theta<1/2$, the following a posteriori err

Figures (2)

  • Figure 1: Example \ref{['example3']}: Time step-size trajectory (top row) and the error (bottom row) with $Tol=0.9$, $k_0=1/60$, and $k_{\max}=0.1$
  • Figure 2: Example \ref{['example4']}: Time step-size trajectory (top row) and the error (bottom row) with $Tol=0.1$, $k_0=0.1$, and $k_{\max}=0.12$

Theorems & Definitions (15)

  • Theorem 2.1
  • Proof 1
  • Remark 2.2
  • Remark 3.1: Optimality of the residual $\hat{R}$
  • Theorem 3.2
  • Proof 2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • ...and 5 more