Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral Deferred Corrections in the framework of Runge-Kutta methods

Eugen Bronasco, Joscha Fregin, Daniel Ruprecht, Gilles Vilmart

Abstract

We interpret a wide range of flavors of Spectral Deferred Corrections (SDC) as Runge-Kutta methods (RKM). Using Butcher series, we show that the considered class of SDC methods achieve at least order p after p iterations compared to the underlying RKM, independently of the error discretisation chosen and the choice of nodes. For all collocation RKM, we analyse the phenomenon of order jumps in SDC iterations, where the order is increased by two at each iteration. We prove that it can be obtained by using appropriate inconsistent, implicit, parallelisable error discretisations. We also investigate the stability properties of the new SDC methods which can in general reduce to that of explicit RKM, but it can be improved by suitable combinations of error discretisations. We confirm the convergence analysis with numerical experiments and we apply relaxation RKM to derive SDC variants that conserve quadratic invariants.

Spectral Deferred Corrections in the framework of Runge-Kutta methods

Abstract

We interpret a wide range of flavors of Spectral Deferred Corrections (SDC) as Runge-Kutta methods (RKM). Using Butcher series, we show that the considered class of SDC methods achieve at least order p after p iterations compared to the underlying RKM, independently of the error discretisation chosen and the choice of nodes. For all collocation RKM, we analyse the phenomenon of order jumps in SDC iterations, where the order is increased by two at each iteration. We prove that it can be obtained by using appropriate inconsistent, implicit, parallelisable error discretisations. We also investigate the stability properties of the new SDC methods which can in general reduce to that of explicit RKM, but it can be improved by suitable combinations of error discretisations. We confirm the convergence analysis with numerical experiments and we apply relaxation RKM to derive SDC variants that conserve quadratic invariants.

Paper Structure

This paper contains 16 sections, 12 theorems, 91 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Related work on the theory of SDC.
  3. Order jumps in SDC.
  4. Spectral deferred corrections
  5. Butcher series for SDC
  6. Convergence analysis of SDC methods
  7. Convergence order of SDC
  8. Analysis of order jumps
  9. Order jumps for linear problems
  10. Order jumps for non-linear problems
  11. Stability and numerical experiments
  12. Stability and convergence in the limit $k \to \infty$
  13. Stability and convergence for a fixed number $k$ of iterations
  14. An S-conservative SDC method
  15. Summary
  16. ...and 1 more sections

Key Result

Proposition 1

Let $(\tilde{\bm{A}}, \bm{b}, \bm{c}) = (\bm{A}_\Delta^0, \ldots, \bm{A}^K_\Delta, \bm{A}, \bm{b}, \bm{c})$ define an implicit SDC method. If $\tilde{\bm{A}}$ is not singular and satisfies $a_{si} = b_i$, the method is stiffly accurate, then $\lim_{z \to \infty} R(z) = 0$. $\blacktriangleleft$$\blac

Figures (13)

  • Figure 1: Comparison of the size and height of trees.
  • Figure 2: Simplifying assumptions and corresponding order of collocation methods with $s$ stages.
  • Figure 3: $M=3$ Radau IIA nodes were used for the figure. Left: $\bar{\rho}_k(\tilde{\bm{B}}, \Vert \cdot \Vert_\infty) \leq 1$ inside enclosed contour lines for $k=10^1, 10^2, 10^3$ and $(\bm{A}_{\Delta_{\mathtt{J}}}^k, \bm{A}, \bm{b}, \bm{c})$ (blue solid lines) as well as Picard iterations $(\bm{A}_{\Delta_{\mathtt{P}}}^k, \bm{A}, \bm{b}, \bm{c})$ (dotted red lines). Right: Stability domain for $(\bm{A}_{\Delta_{\mathtt{J}}}^k, \bm{A}, \bm{b},\bm{c})$ (blue solid lines) and $(\bm{A}_{\Delta_{\mathtt{P}}}^k, \bm{A}, \bm{b}, \bm{c})$ (red dotted lines) after $k=10^1, 10^2, 10^3$ iterations. Stability domain is inside enclosed contour lines. For both plots the contours of $(\bm{A}_{\Delta_{\mathtt{J}}}^k, \bm{A}, \bm{b}, \bm{c})$ approach the contours of $(\bm{A}_{\Delta_{\mathtt{P}}}^k, \bm{A}, \bm{b}, \bm{c})$ with increasing iteration count $k$.
  • Figure 4: SDC method $(\bm{A}_{\Delta_\text{J}}, \bm{A}_{\Delta_\text{J}}, \bm{A}_{\Delta_\text{J}}, \mathop{\mathrm{diag}}\nolimits(\bm{c}), \mathop{\mathrm{diag}}\nolimits(\bm{c}/3), \bm{A}, \bm{b}, \bm{c})$ with $M=5$ and RadauIIA nodes. Left: $\bar{\rho}_5(\tilde{\bm{B}}, \Vert \cdot \Vert_\infty)$ along the real (dashed, blue) and imaginary (dash dotted, red) axes. Right: $|R(z)|$ along the real (dashed, blue) and imaginary (dash dotted, red) axes. For both plots the graphs approach zero with increasing $z$, which is to be expected by proposition \ref{['prop:zeroerror']}.
  • Figure 5: Numerical convergence of SDC with $M=6$ Radau IIA nodes and the EED introduced in Theorem \ref{['thm:order_jump']}. The last node was used as the solution in each timestep. The grey dashed lines serve as a guide to the eye with slopes of 2, 4, 6, 8 and 10. The method shows the expected 2nd, 4th, 6th, 8th, and 10th order convergence for $k=1,2,3,4,5$ iterations (see \ref{['thm:order_jump']}). Left: Dahlquist test equation with $t_0=0$, $t_{\text{end}}=1$, $\lambda = -1$ and $u_0=1$. The data point of $k=5$ close to $\Delta t = 10^{-1}$ that appears to be missing is below the scale shown. Within this area the error of the method with $k=5$ is close to machine precision, which is why slope appears to be 0. Right: Euler rigid body equations. The implementation using SciPy fsolve limits the precision to $\sim 10^{-14}$. See Fig. \ref{['table:JUMPERRADAU']}.
  • ...and 8 more figures

Theorems & Definitions (35)

  • Proposition 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1: rooted tree
  • Definition 2: tree functions
  • Definition 3: Butcher series
  • Definition 4: SDC order
  • Definition 5: tree height
  • ...and 25 more