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Stability of step size control based on a posteriori error estimates

Hendrik Ranocha, Jan Giesselmann

TL;DR

This work compares the stability properties of residual-based estimators and classical error estimators based on an embedded Runge-Kutta method both analytically and in numerical experiments and demonstrates that a standard I controller is unstable while more advanced PI and PID controllers can be designed to be stable.

Abstract

A posteriori error estimates based on residuals can be used for reliable error control of numerical methods. Here, we consider them in the context of ordinary differential equations and Runge-Kutta methods. In particular, we take the approach of Dedner & Giesselmann (2016) and investigate it when used to select the time step size. We focus on step size control stability when combined with explicit Runge-Kutta methods and demonstrate that a standard I controller is unstable while more advanced PI and PID controllers can be designed to be stable. We compare the stability properties of residual-based estimators and classical error estimators based on an embedded Runge-Kutta method both analytically and in numerical experiments.

Stability of step size control based on a posteriori error estimates

TL;DR

This work compares the stability properties of residual-based estimators and classical error estimators based on an embedded Runge-Kutta method both analytically and in numerical experiments and demonstrates that a standard I controller is unstable while more advanced PI and PID controllers can be designed to be stable.

Abstract

A posteriori error estimates based on residuals can be used for reliable error control of numerical methods. Here, we consider them in the context of ordinary differential equations and Runge-Kutta methods. In particular, we take the approach of Dedner & Giesselmann (2016) and investigate it when used to select the time step size. We focus on step size control stability when combined with explicit Runge-Kutta methods and demonstrate that a standard I controller is unstable while more advanced PI and PID controllers can be designed to be stable. We compare the stability properties of residual-based estimators and classical error estimators based on an embedded Runge-Kutta method both analytically and in numerical experiments.
Paper Structure (18 sections, 1 theorem, 63 equations, 10 figures, 2 tables)

This paper contains 18 sections, 1 theorem, 63 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Basic ideas of step size control and a posteriori error estimates
  3. Energy based a posteriori error estimates
  4. Error per step versus error per unit step
  5. Step size control stability
  6. Explicit Euler method
  7. Second-order, two-stage methods
  8. Third-order, three-stage methods
  9. Left-biased cubic Hermite interpolation
  10. Central cubic Hermite interpolation
  11. Numerical experiments
  12. A nonlinear ODE
  13. Euler equations of a rigid body
  14. 1D Benjamin-Bona-Mahony equation
  15. Reliability of the global error estimate
  16. ...and 3 more sections

Key Result

lemma 2.1

Let $u$ be a solution of eq:ode and let $\widehat{u}$ solve eq:res. Then, for all $0 \leq t \leq T$ where $L$ is such that eq:osL holds. If $\widehat{u}$ is the reconstruction of a numerical solution satisfying $\widehat{u}(t^n) = u^n$, then as an immediate consequence for all $0 \leq t^n \leq T$

Figures (10)

  • Figure 1: Spectral radius of the Jacobian $J$ for error estimates based on the residual and an embedded Euler method for explicit second-order, two-stage Runge-Kutta methods. The Jacobian is evaluated at $z = r \mathrm{e}^{\mathrm{i} \varphi}$ where the radius $r$ is chosen such that $z$ is on the boundary of the stability region of the (main) method.
  • Figure 2: Spectral radius of the Jacobian $J$ for error estimates based on the $L^1$ residual and the second-order embedded method for the third-order method of Bogacki and Shampine bogacki1989a32 with left-biased cubic Hermite interpolation. The Jacobian is evaluated at $z = r \mathrm{e}^{\mathrm{i} \varphi}$ where the radius $r$ is chosen such that $z$ is on the boundary of the stability region of the (main) method.
  • Figure 3: Spectral radius of the Jacobian $J$ for error estimates based on the $L^1$ residual and the second-order embedded method for the third-order method of Bogacki and Shampine bogacki1989a32 with central cubic Hermite interpolation. The Jacobian is evaluated at $z = r \mathrm{e}^{\mathrm{i} \varphi}$ where the radius $r$ is chosen such that $z$ is on the boundary of the stability region of the (main) method.
  • Figure 4: Spectral radius of the Jacobian $J$ for error estimates based on the $L^2$ residual and the second-order embedded method for the third-order method of Bogacki and Shampine bogacki1989a32 with central cubic Hermite interpolation. The Jacobian is evaluated at $z = r \mathrm{e}^{\mathrm{i} \varphi}$ where the radius $r$ is chosen such that $z$ is on the boundary of the stability region of the (main) method.
  • Figure 5: Number of rejected steps for the nonlinear ODE \ref{['eq:krogh']} using the the third-order method of Bogacki and Shampine bogacki1989a32 with tolerance $\tau = 10^{-4}$.
  • ...and 5 more figures

Theorems & Definitions (2)

  • lemma 2.1
  • proof : Proof of Lemma \ref{['lem:estimate']}