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Adaptive Step Selection for a Filtered Implicit Method

Stephen M. McGovern

TL;DR

This work develops an adaptive time-stepping framework for two higher-order filtered implicit Euler methods by introducing variable-step pre- and post-filters. It derives explicit adaptive coefficients, with the pre-filter coefficient $\\alpha_n = \\frac{k_n^2}{k_{n-1}k_{n-2}}$ for second order and a ratio form for the post-filter coefficient $\\beta_n$ to achieve third order, yielding the embedded Filtered-IE23 pair. An error estimator $EST = |y^{3rd}_{n+1} - y^{2nd}_{n+1}|$ drives a halving/doubling controller to adapt the time step. Numerical experiments across model, quasi-periodic, nonautonomous, and stiff problems demonstrate accurate, stable performance and competitive efficiency relative to constant-step higher-order methods, with open-source code provided at the project repository.

Abstract

Pre-filtering and post-filtering steps can be added to many of the traditional numerical methods to generate new, higher order methods with strong stability properties. Presented in this paper are a variable step pre-filter and post-filter that allow adaptive time stepping for a filtered method based on Implicit Euler.

Adaptive Step Selection for a Filtered Implicit Method

TL;DR

This work develops an adaptive time-stepping framework for two higher-order filtered implicit Euler methods by introducing variable-step pre- and post-filters. It derives explicit adaptive coefficients, with the pre-filter coefficient for second order and a ratio form for the post-filter coefficient to achieve third order, yielding the embedded Filtered-IE23 pair. An error estimator drives a halving/doubling controller to adapt the time step. Numerical experiments across model, quasi-periodic, nonautonomous, and stiff problems demonstrate accurate, stable performance and competitive efficiency relative to constant-step higher-order methods, with open-source code provided at the project repository.

Abstract

Pre-filtering and post-filtering steps can be added to many of the traditional numerical methods to generate new, higher order methods with strong stability properties. Presented in this paper are a variable step pre-filter and post-filter that allow adaptive time stepping for a filtered method based on Implicit Euler.

Paper Structure

This paper contains 17 sections, 3 theorems, 32 equations, 7 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Main Results
  4. Adaptive Step Pre-filter
  5. Adaptive Step Post-filter
  6. Algorithm
  7. Numerical results
  8. Model Problem
  9. Quasi-Periodic Oscillation ODE
  10. Model Problem Analog
  11. van der Pol Oscillator
  12. Conclusions
  13. Data Availability Statement
  14. Competing Interest Declaration
  15. Ethical Statement
  16. ...and 2 more sections

Key Result

Proposition 1

The value for the pre-filter $\alpha_n$ which gives a second order adaptive method after completing Step 1) and Step 2) is

Figures (7)

  • Figure 1: Quasi-Periodic Oscillation ODE Solution
  • Figure 2: Constant Step Methods set to $\Delta t = 0.1$
  • Figure 3: Constant Step Methods set to $\Delta t = 0.05$
  • Figure 4: Model ODE Analog Solution $\gamma=5$
  • Figure 5: Model ODE Analog Solution $\gamma=6$
  • ...and 2 more figures

Theorems & Definitions (6)

  • Proposition 1
  • Proof 1
  • Proposition 2
  • Proof 2
  • Lemma 3.1
  • Proof 3