Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence and stability of randomized implicit two-stage Runge-Kutta schemes

Tomasz Bochacik, Paweł Przybyłowicz

TL;DR

This work analyzes two randomized implicit two-stage Runge-Kutta schemes for initial value problems, introducing a uniform randomization of the intermediate slope. The authors prove that randomness increases the convergence rate by $ frac{1}{2}$, achieving an $L^p$ error of order $h^{ ho+ rac{1}{2}}$, and they establish probabilistic stability properties via Dahlquist’s test equation: the schemes are asymptotically A-stable and stable in probability, but not mean-square A-stable. The mean-square stability region is shown to be bounded, while the asymptotic and probability-based stability regions coincide with the left half-plane. Numerical experiments on a stiff problem corroborate the theoretical results, showing robust behavior of the randomized implicit schemes and superior stability compared to explicit methods, especially for small step sizes.

Abstract

We randomize the implicit two-stage Runge-Kutta scheme in order to improve the rate of convergence (with respect to a deterministic scheme) and stability of the approximate solution (with respect to the solution generated by the explicit scheme). For stability analysis, we use Dahlquist's concept of A-stability, adopted to randomized schemes by considering three notions of stability: asymptotic, mean-square, and in probability. The randomized implicit RK2 scheme proves to be A-stable asymptotically and in probability but not in the mean-square sense.

Convergence and stability of randomized implicit two-stage Runge-Kutta schemes

TL;DR

This work analyzes two randomized implicit two-stage Runge-Kutta schemes for initial value problems, introducing a uniform randomization of the intermediate slope. The authors prove that randomness increases the convergence rate by , achieving an error of order , and they establish probabilistic stability properties via Dahlquist’s test equation: the schemes are asymptotically A-stable and stable in probability, but not mean-square A-stable. The mean-square stability region is shown to be bounded, while the asymptotic and probability-based stability regions coincide with the left half-plane. Numerical experiments on a stiff problem corroborate the theoretical results, showing robust behavior of the randomized implicit schemes and superior stability compared to explicit methods, especially for small step sizes.

Abstract

We randomize the implicit two-stage Runge-Kutta scheme in order to improve the rate of convergence (with respect to a deterministic scheme) and stability of the approximate solution (with respect to the solution generated by the explicit scheme). For stability analysis, we use Dahlquist's concept of A-stability, adopted to randomized schemes by considering three notions of stability: asymptotic, mean-square, and in probability. The randomized implicit RK2 scheme proves to be A-stable asymptotically and in probability but not in the mean-square sense.
Paper Structure (13 sections, 4 theorems, 60 equations, 4 figures)

This paper contains 13 sections, 4 theorems, 60 equations, 4 figures.

Table of Contents

  1. Preliminaries
  2. Initial problem
  3. Schemes
  4. Error bounds
  5. Class of initial value problems
  6. Existence of the solution
  7. Results on the $L^p(\Omega)$-error
  8. Probabilistic stability
  9. Test equation
  10. Regions of mean-square stability
  11. Regions of asymptotic stability and stability in probability
  12. Numerical experiments
  13. Conclusions and future work

Key Result

Lemma 1

Let $\left(\eta,f\right)\in F^\varrho$ and let us assume that $Lh<1$. Then for every $j\in\{1,\ldots,n\}$ there exists a solution $V_\tau^j$ to the equation cf. eq:S1. This solution is unique (up to a null event) and $\mathcal{F}_j$-measurable.

Figures (4)

  • Figure 1: Geometric interpretation of moduli that appeared in \ref{['eq:MS:C-']}.
  • Figure 2: Contours of the stability regions for the RK2 scheme: deterministic explicit, randomized explicit (asymptotic) $\mathcal{R}^e_{AS}$, randomized explicit (mean-square) $\mathcal{R}^e_{MS}$, and randomized implicit (mean-square) $\mathcal{R}_{MS}$.
  • Figure 3: Numerical solution of \ref{['eq:num-test']} obtained via schemes \ref{['eq:detS1']} and \ref{['eq:S1']} for step size $h\in\bigl\{\frac{1}{2},\frac{1}{4},\frac{1}{8}\bigr\}$.
  • Figure 4: Numerical solution of \ref{['eq:num-test']} obtained via schemes \ref{['eq:detS2']} and \ref{['eq:S2']} for step size $h\in\bigl\{\frac{1}{2},\frac{1}{4},\frac{1}{8}\bigr\}$.

Theorems & Definitions (16)

  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • proof
  • proof
  • proof
  • ...and 6 more