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Stable corrections for perturbed diagonally implicit Runge--Kutta methods

John Driscoll, Sigal Gottlieb, Zachary J. Grant, César Herrera, Tej Sai Kakumanu, Michael H. Sawicki, Monica Stephens

Abstract

A mixed accuracy framework for Runge--Kutta methods presented in Grant [JSC 2022] and applied to diagonally implicit Runge--Kutta (DIRK) methods can significantly speed up the computation by replacing the implicit solver by less expensive low accuracy approaches such as lower precision computation of the implicit solve, under-resolved iterative solvers, or simpler, less accurate models for the implicit stages. Understanding the effect of the perturbation errors introduced by the low accuracy computations enables the design of stable and accurate mixed accuracy DIRK methods where the errors from the low-accuracy computation are damped out by multiplication by \dt at multiple points in the simulation, resulting in a more accurate simulation than if low-accuracy was used for all computation. To improve upon this, explicit corrections were previously proposed and analyzed for accuracy, and their performance was tested in related work. Explicit corrections work well when the time-step is sufficiently small, but may introduce instabilities when the time-step is larger. In this work, the stability of the mixed accuracy approach is carefully studied, and used to design novel stabilized correction approaches.

Stable corrections for perturbed diagonally implicit Runge--Kutta methods

Abstract

A mixed accuracy framework for Runge--Kutta methods presented in Grant [JSC 2022] and applied to diagonally implicit Runge--Kutta (DIRK) methods can significantly speed up the computation by replacing the implicit solver by less expensive low accuracy approaches such as lower precision computation of the implicit solve, under-resolved iterative solvers, or simpler, less accurate models for the implicit stages. Understanding the effect of the perturbation errors introduced by the low accuracy computations enables the design of stable and accurate mixed accuracy DIRK methods where the errors from the low-accuracy computation are damped out by multiplication by \dt at multiple points in the simulation, resulting in a more accurate simulation than if low-accuracy was used for all computation. To improve upon this, explicit corrections were previously proposed and analyzed for accuracy, and their performance was tested in related work. Explicit corrections work well when the time-step is sufficiently small, but may introduce instabilities when the time-step is larger. In this work, the stability of the mixed accuracy approach is carefully studied, and used to design novel stabilized correction approaches.

Paper Structure

This paper contains 25 sections, 4 theorems, 108 equations, 12 figures.

Table of Contents

  1. Overview
  2. Accuracy and stability analysis of perturbed methods
  3. Bounding the internal stage errors
  4. Implicit midpoint rule
  5. The two stage third order SDIRK method
  6. General DIRK method
  7. Understanding the perturbation errors: a numerical study using Burgers' equation
  8. Linearization & perturbation
  9. Mixed precision implementation with an iterative solver
  10. Stability and accuracy of corrections
  11. Stabilizing the explicit correction approach
  12. Analyzing the stabilized corrections as a time-stepping method
  13. Defining the stabilization matrix $\Phi$
  14. Static stabilization
  15. Dynamic stabilizations
  16. ...and 10 more sections

Key Result

Lemma 1

Given a differential equation of the form ODE that is evolved forward with the method MPDIRK using the function $f_{\varepsilon}$ where If the coefficients of MPDIRK satisfy the conditions Coefficients, then the growth of the errors resulting from $h$ is bounded by:

Figures (12)

  • Figure 1: The time evolution of $\|h(y^{i})\|_\infty$ for the linearized Burgers' equation with initial condition $\frac{1}{2} + \frac{1}{4} \sin(x)$ evolved to time $T_f=3.5$. Left: mixed accuracy SDIRK2 \ref{['MPIMR']}; Middle: mixed accuracy SDIRK3 \ref{['MPSDIRK3']}; Right: mixed accuracy SDIRK4 \ref{['MPSDIRK4']}. Red lines are $\Delta t = 0.1$, Green lines $\Delta t = 0.01$, Blue lines $\Delta t = 0.001$. The dotted lines are for $N_x = 30$, solid lines are $N_x=50$, dashed lines $N_x=250$.
  • Figure 1: Correction approaches for the time-evolution of the inviscid Burgers' with $N_x=50$ (left), $N_x=250$ (middle), and $N_x=450$ (right). We evolve this to final time $T_f= 3.5$ using SDIRK2 in blue, SDIRK3 in red, and SDIRK4 in green. We use no corrections (solid lines), explicit corrections (dashed line), and the static stabilized correction $\Phi_J$ (dotted line). The method with no perturbation has square markers, the perturbed method with $\epsilon = 10^{-4}$ has round markers.
  • Figure 2: The final time maximum norm errors of the linearized and perturbed Burgers' equation, compared to a reference solution. In blue we have the mixed accuracy SDIRK2 \ref{['MPIMR']}, in red the mixed accuracy SDIRK3 \ref{['MPSDIRK3']}, and in green the mixed accuracy SDIRK4 \ref{['MPSDIRK4']}. Left: $N_x = 50$; Middle $N_x = 250$; Right $N_x = 450$. Solid lines are only linearized, but not perturbed; Dashed lines have a perturbation of $\epsilon = 10^{-6}$; Dash-dot lines have a perturbation of $\epsilon = 10^{-4}$; Dotted lines have a perturbation of $\epsilon = 10^{-2}$.
  • Figure 2: Mixed precision Burgers' equation with $N_x = 200$ spatial points. SDIRK2 \ref{['MPIMR']} in blue; SDIRK3 \ref{['MPSDIRK3']} in red; SDIRK4 \ref{['MPSDIRK4']} in green. Top Left: No corrections. Top Right: two explicit corrections. Bottom Left: two corrections with $\Phi_{EIN}$ stabilization. Bottom Right: two corrections with $\Phi_J$ stabilization.
  • Figure 3: Time evolution of $\left| h \right|_\infty$ for mixed precision Burgers' equation for double/single (64/32) on top and quad/double (128/64) on bottom. Blue: SDIRK2; red: SDIRK3; green: SDIRK4. Dotted: $\Delta t = 10^{-2}$; dash-dotted: $\Delta t = 10^{-3}$; dash: $\Delta t = 10^{-4}$. Left: $N_x = 50$; center: $N_x = 100$; right: $Nx = 200$. Each marker corresponds to a different stage.
  • ...and 7 more figures

Theorems & Definitions (9)

  • Lemma 1
  • Proof 1
  • Proposition 1
  • Proof 2
  • Lemma 2
  • Proof 3
  • Theorem 1
  • Proof 4
  • Remark 1