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Gautschi-type and implicit-explicit integrators for constrained wave equations

R. Altmann, B. Dörich, C. Zimmer

TL;DR

The paper addresses time integration for semi-linear second-order PDAEs with linear constraints, formulating the problem in a Hilbert-space framework and exploiting a kernel/complement decomposition to handle constraints. It develops and analyzes two approaches: an implicit–explicit Crank–Nicolson (IMEX CN) scheme and a Gautschi-type exponential integrator, both achieving second-order accuracy. The IMEX CN method requires solving saddle-point problems at each time step, while the Gautschi-type method operates on the constrained kernel and uses cos/sinc operator functions, with practical implementation via Krylov subspaces and saddle-point solvers. Numerical experiments on constrained wave equations with kinetic boundary conditions verify the theoretical convergence and illustrate efficient practical performance, highlighting the methods' suitability for constrained wave dynamics and their potential for extension to more general exponential integrators.

Abstract

This paper deals with the construction and analysis of two integrators for (semi-linear) second-order partial differential-algebraic equations of semi-explicit type. More precisely, we consider an implicit-explicit Crank-Nicolson scheme as well as an exponential integrator of Gautschi type. For this, well-known wave integrators for unconstrained systems are combined with techniques known from the field of differential-algebraic equations. This results in efficient time stepping schemes that are provable of second order. Moreover, we discuss the practical implementation of the Gautschi-type method, which involves the solution of certain saddle point problems. The theoretical results are verified by a numerical experiment for the wave equation with kinetic boundary conditions.

Gautschi-type and implicit-explicit integrators for constrained wave equations

TL;DR

The paper addresses time integration for semi-linear second-order PDAEs with linear constraints, formulating the problem in a Hilbert-space framework and exploiting a kernel/complement decomposition to handle constraints. It develops and analyzes two approaches: an implicit–explicit Crank–Nicolson (IMEX CN) scheme and a Gautschi-type exponential integrator, both achieving second-order accuracy. The IMEX CN method requires solving saddle-point problems at each time step, while the Gautschi-type method operates on the constrained kernel and uses cos/sinc operator functions, with practical implementation via Krylov subspaces and saddle-point solvers. Numerical experiments on constrained wave equations with kinetic boundary conditions verify the theoretical convergence and illustrate efficient practical performance, highlighting the methods' suitability for constrained wave dynamics and their potential for extension to more general exponential integrators.

Abstract

This paper deals with the construction and analysis of two integrators for (semi-linear) second-order partial differential-algebraic equations of semi-explicit type. More precisely, we consider an implicit-explicit Crank-Nicolson scheme as well as an exponential integrator of Gautschi type. For this, well-known wave integrators for unconstrained systems are combined with techniques known from the field of differential-algebraic equations. This results in efficient time stepping schemes that are provable of second order. Moreover, we discuss the practical implementation of the Gautschi-type method, which involves the solution of certain saddle point problems. The theoretical results are verified by a numerical experiment for the wave equation with kinetic boundary conditions.

Paper Structure

This paper contains 16 sections, 12 theorems, 132 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Wave-type Systems with Linear Constraints
  3. Problem formulation
  4. Existence of solutions
  5. An implicit--explicit Crank--Nicolson Scheme
  6. Review of the unconstrained case
  7. An implicit--explicit scheme of second order
  8. Convergence analysis
  9. A Gautschi-type Integrator
  10. Review of the unconstrained case
  11. An implicit solution formula
  12. A Gautschi-type integration scheme
  13. Convergence analysis
  14. Practical Implementation
  15. Numerical Example
  16. ...and 1 more sections

Key Result

Lemma 2.10

Let Assumption assD hold and consider $\mathcal{A} = \mathcal{A}_1+\mathcal{A}_2$ with $\mathcal{A}_1 \in \mathcal{L}(\mathcal{V},\mathcal{V}^*)$ being elliptic with constant $\mu_1$ and self-adjoint and $\mathcal{A}_2 \in \mathcal{L}(\mathcal{V},\mathcal{H}^*)$ with continuity constant $C_2$. Furth with initial conditions $u(0) = u^0$ and $\dot u(0) = w^0$ has a unique solution This solution sat

Figures (1)

  • Figure 5.1: Convergence history for different time stepping schemes. Plot shows error in $u$, measured in the $L^\infty(0,T;L^2(\Omega))$-norm. The gray lines indicate orders $1$ (dashed) and $2$ (dotted).

Theorems & Definitions (34)

  • Remark 2.4
  • Remark 2.5
  • Remark 2.7
  • Example 2.8
  • Example 2.9
  • Lemma 2.10: Existence of solutions, linear and unconstrained case
  • proof
  • Lemma 2.11: Existence of solutions, linear PDAE case
  • proof
  • Theorem 2.12: Existence of solutions
  • ...and 24 more