An unconditionally stable space-time isogeometric method for the acoustic wave equation

TL;DR

A space--time variational formulation that is obtained by adding a non-consistent penalty term of order $2p+2$ to the bilinear form coming from integration by parts, which, when discretized with tensor-product spline spaces with maximal regularity in time, is unconditionally stable.

Abstract

We study space--time isogeometric discretizations of the linear acoustic wave equation that use splines of arbitrary degree p, both in space and time. We propose a space--time variational formulation that is obtained by adding a non-consistent penalty term of order 2p+2 to the bilinear form coming from integration by parts. This formulation, when discretized with tensor-product spline spaces with maximal regularity in time, is unconditionally stable: the mesh size in time is not constrained by the mesh size in space. We give extensive numerical evidence for the good stability, approximation, dissipation and dispersion properties of the stabilized isogeometric formulation, comparing against stabilized finite element schemes, for a range of wave propagation problems with constant and variable wave speed.

Figures (21)

• Figure 1: Relative errors of the IGA-Stab method \ref{['equivstabiga']} with $h_s=2^{-7}$ and $h_t=5h_s$, plotted against the stabilization parameter $\delta$. The exact solution is defined in \ref{['u1_ex']}. The markers correspond to $\delta=10^{-p}$.
• Figure 2: Relative errors of the IGA-Stab method \ref{['equivstabiga']} plotted against the ratio $h_t/h_s$ with fixed $h_t=0.1562$. The exact solution is defined in \ref{['u1_ex']}. We observe that the method does not require any CFL condition: reducing the space mesh size, the error remains bounded.
• Figure 3: Comparison between relative errors of the IGA-Stab method \ref{['equivstabiga']}, with splines of maximal regularity (continuous lines), and relative errors of the FEM-Stab method \ref{['stabZank']}, with $C^0$ splines (dashed lines). Top: space--time errors plotted against the total number of DOFs $N_{\mathrm{dof}}$, with $h_t = 5 h_s$. Bottom: final-time errors plotted against the spline degree $p$, with $N_{\mathrm{dof}}=7\,080$ and $h_t \approx 5 h_s$ for both the stabilizations and all the spline degrees.
• Figure 4: Comparison between relative errors of the IGA-Stab method \ref{['equivstabiga']} (continuous lines) and relative errors of the FEM-Stab method \ref{['stabZank']} (dashed lines) plotted against the number of space DOFs per wavelength $N_{\mathrm{dof}}/\sharp \lambda$, at different wavenumbers $k$. $L^2(Q)$ norms are shown on the left, $H^1(Q)$ seminorms on the right. Rows 1 to 4 correspond to $p=1$ to $p=4$. The exact solution is defined in \ref{['u2_ex']}.
• Figure 5: Relative errors of the IGA-Stab method \ref{['equivstabiga']} plotted against the time mesh-size $h_t \approx h_s$ for the scattering problem \ref{['ex:scattering']}.

Theorems & Definitions (2)

• Remark 1
• Remark 2