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Unconditionally stable space-time isogeometric discretization for the wave equation in Hamiltonian formulation

Matteo Ferrari, Sara Fraschini, Gabriele Loli, Ilaria Perugia

Abstract

We consider a family of conforming space-time discretizations for the wave equation based on a first-order-in-time formulation employing maximal regularity splines. In contrast with second-order-in-time formulations, which require a CFL condition to guarantee stability, the methods we consider here are unconditionally stable without the need for stabilization terms. Along the lines of the work by M. Ferrari and S. Fraschini (2024), we address the stability analysis by studying the properties of the condition number of a family of matrices associated with the time discretization. Numerical tests validate the performance of the method.

Unconditionally stable space-time isogeometric discretization for the wave equation in Hamiltonian formulation

Abstract

We consider a family of conforming space-time discretizations for the wave equation based on a first-order-in-time formulation employing maximal regularity splines. In contrast with second-order-in-time formulations, which require a CFL condition to guarantee stability, the methods we consider here are unconditionally stable without the need for stabilization terms. Along the lines of the work by M. Ferrari and S. Fraschini (2024), we address the stability analysis by studying the properties of the condition number of a family of matrices associated with the time discretization. Numerical tests validate the performance of the method.

Paper Structure

This paper contains 22 sections, 21 theorems, 164 equations, 14 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. First-order-in-time variational formulation and temporal discretization with maximal regularity splines
  3. Associated ODE and matricial formulation
  4. Conditioning of the involved matrices
  5. Conditioning of Toeplitz band matrices with symmetry
  6. Conditioning of Toeplitz band matrices with skew-symmetry
  7. Conditioning of the Schur complements
  8. Why equal trial and test spaces fail
  9. Numerical experiments
  10. Efficient implementation
  11. Numerical tests
  12. Example 1. Unconditional stability and optimal convergence rates
  13. Example 2. Highly oscillatory solutions
  14. Example 3. More general boundary conditions: a scattering problem
  15. Example 4. Singular solution
  16. ...and 7 more sections

Key Result

Proposition 2.3

Let $p \ge 1$ and let $\mathbf{B}_h^p$ and $\mathbf{C}_h^p$ be defined in eq:7. Then, the following properties are valid:

Figures (14)

  • Figure 1: Spectral condition numbers of the Schur complements in \ref{['eq:53']} in semi-logarithmic scale, with $n=1000$ by varying $\rho \in [2.5,5.5]$, with $p \in \{1,2,3\}$.
  • Figure 2: Spectral condition numbers of the Schur complements in \ref{['eq:53']} in semi-logarithmic scale, with $n=1000$ by varying $\rho \in [5.5,7]$, with $p \in \{4,5,6\}$.
  • Figure 3: Example 1. Relative errors plotted against the ratio $h_t/h_x$ with fixed $h_t=0.1562$. The exact solution is defined in \ref{['u1_ex']}.
  • Figure 4: Example 1. First row: relative errors between the exact position $U$ and the discrete one $U_{\boldsymbol{h}}^p$ provided by the unconditionally stable method \ref{['eq:4']} (continuous lines ), the unconditionally stable method in FrenchPeterson1996 (dashed lines ), and the stabilized method devised in FraschiniLoliMoiolaSangalli2023 (dash-dotted lines $\cdot$ ). Second row: relative errors between the exact velocity $V$ and the discrete one $V_{\boldsymbol{h}}^p$ provided by method \ref{['eq:4']} (continuous lines ) and FrenchPeterson1996 (dashed lines ). The errors are plotted against the total number of DOFs $N_{\mathrm{dof}}$, and the mesh sizes satisfy $h_t = 5 h_x$.
  • Figure 5: Example 2. Relative errors of \ref{['eq:4']} plotted against the number of space DOFs per wave length $N_{\mathrm{dof}}/\sharp \lambda$, at different wave numbers $k$. $L^2$ norms are shown on the left, $H^1$ seminorms on the right. Rows 1 to 4 correspond to $p=1$ to $p=4$. The exact solution is defined in \ref{['u2_ex']}.
  • ...and 9 more figures

Theorems & Definitions (53)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.5: Uniqueness at the continuous level
  • Remark 2.6: Uniqueness at the discrete level for small $\mu$
  • Remark 2.7
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • ...and 43 more