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Numerical study of Galerkin-collocation approximation in time for the wave equation

Mathias Anselmann, Markus Bause

TL;DR

This work develops and numerically assesses Galerkin–collocation time discretizations for the wave equation within a space–time finite element framework. By marrying variational time discretization with collocation constraints, the authors achieve higher temporal regularity and reduced linear algebra compared to standard continuous schemes. They introduce two schemes, GCC$^1(3)$ and GCC$^2(5)$, detailing their fully discrete algebraic forms, solver strategies (condensed CG and GMRES with AMG), and convergence behavior, including a SHM-inspired test comparing against cGP(2). The results show optimal space–time convergence and substantial computational savings due to smaller, well-conditioned systems, suggesting favorable applicability to multi-physics problems requiring accurate time integration.

Abstract

The elucidation of many physical problems in science and engineering is subject to the accurate numerical modelling of complex wave propagation phenomena. Over the last decades, high-order numerical approximation for partial differential equations has become a well-established tool. Here we propose and study numerically the implicit approximation in time of wave equations by a Galerkin--collocation approach that relies on a higher order space-time finite element approach. The conceptual basis is the establishment of a direct connection between the Galerkin method for the time discretization and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs provided by the latter in terms of less complex linear algebraic systems. For the fully discrete solution, higher order regularity in time is further ensured which can be advantageous in the discretization of multi-physics systems. The accuracy and efficiency of the variational collocation approach is carefully studied by numerical experiments.

Numerical study of Galerkin-collocation approximation in time for the wave equation

TL;DR

This work develops and numerically assesses Galerkin–collocation time discretizations for the wave equation within a space–time finite element framework. By marrying variational time discretization with collocation constraints, the authors achieve higher temporal regularity and reduced linear algebra compared to standard continuous schemes. They introduce two schemes, GCC and GCC, detailing their fully discrete algebraic forms, solver strategies (condensed CG and GMRES with AMG), and convergence behavior, including a SHM-inspired test comparing against cGP(2). The results show optimal space–time convergence and substantial computational savings due to smaller, well-conditioned systems, suggesting favorable applicability to multi-physics problems requiring accurate time integration.

Abstract

The elucidation of many physical problems in science and engineering is subject to the accurate numerical modelling of complex wave propagation phenomena. Over the last decades, high-order numerical approximation for partial differential equations has become a well-established tool. Here we propose and study numerically the implicit approximation in time of wave equations by a Galerkin--collocation approach that relies on a higher order space-time finite element approach. The conceptual basis is the establishment of a direct connection between the Galerkin method for the time discretization and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs provided by the latter in terms of less complex linear algebraic systems. For the fully discrete solution, higher order regularity in time is further ensured which can be advantageous in the discretization of multi-physics systems. The accuracy and efficiency of the variational collocation approach is carefully studied by numerical experiments.

Paper Structure

This paper contains 13 sections, 1 theorem, 45 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Mathematical problem and notation
  3. Galerkin-collocation schemes
  4. Galerkin--collocation GCC1(3)
  5. Fully discrete system
  6. Solver technology
  7. 1. Approach: Condensing the linear system
  8. 2. Approach: Solving the non-symmetric system
  9. Numerical convergence tests
  10. Test case of structural health monitoring
  11. Galerkin--collocation GCC2(5)
  12. Fully discrete system
  13. Iterative solver and convergence study

Key Result

Theorem 1

Let $l=1$ and $k\geq 3$. For the error $(e^{\; u}, e^{\; v})=(u-u_{\tau,h}, v-v_{\tau,h})$ of the fully discrete scheme GCC$^l(k)$ of Def. Def:GCC there holds that as well as

Figures (4)

  • Figure 1: Concept of structural health monitoring with finite element simulation (scaled displacement field) illustrating the expansion of elastic waves.
  • Figure 2: Preconditioning and solver for the condensed system \ref{['eq:recuced_c1_system']} of GCC$^1(3)$.
  • Figure 3: Test case of structural health monitoring
  • Figure 4: Control quantity \ref{['eq:control_quantity']} for GCC$^1(3)$ (method $C^1$) and cGP(2) (method $C^0$) for different time step sizes.

Theorems & Definitions (4)

  • Remark 1
  • Definition 1: $\boldsymbol{C^l}$--regular in time Galerkin-collocation schemes GCC$\boldsymbol{ {}^l(k)}$
  • Remark 2
  • Theorem 1: Error estimates for ${\boldsymbol{(u_{\tau,h}$, $v_{\tau,h})}}$ of GCC$^1(\boldsymbol k)$