An explicit second order spectral element method for acoustic waves

Abstract

The acoustic wave equation is here discretized by conforming spectral elements in space and by the second order leap-frog method in time. For simplicity, homogeneous boundary conditions are considered. A stability analysis of the resulting method is presented, providing an upper bound for the allowed time step that is proportional to the size of the elements and inversely proportional to the square of their polynomial degree. A convergence analysis is also presented, showing that the convergence error decreases when the time step or the size of the elements decrease or when the polynomial degree increases. Several numerical results illustrating these results are presented.

Local time stepping with the discontinuous Galerkin method for wave propagation in 3D heterogeneous media

“…Figure 4 shows the numerical error against the size of the mesh h ¼ h coarse . We found that, for q ¼ 2 and 4, the convergence rate is of order three, in agreement with the theoretical estimates (Zampieri and Pavarino, 2006;Grote and Schötzau, 2009). Figure 5 displays the recorded signal at one of the receiver's position.…”

Time-harmonic solution for acousto-elastic interaction with controllability and spectral elements

“…We consider the use of a control algorithm to solve the time-harmonic acoustoelastic problem in the domain Ω ⊂ R 2 , which is divided into the solid part Ω s and the fluid part Ω f by the interface Γ i (see Figure 1). Instead of solving directly the time-harmonic equation, we return to the corresponding time dependent equation (see, e.g., [17,10]) and look for time-periodic solution. The convergence is accelerated with a control technique by representing the origi-nal time-harmonic equation as an exact controllability problem [18,19] for the time dependent wave equation…”

“…For space discretization, we use the spectral elements method (see, e.g., [24,17,6,10]), which is based on the weak formulation of the system (1)- (8). That is why we introduce the function spaces…”

“…The time dependent problem is discretized in space domain with a higher-order spectral element method (SEM) and in time domain with second-order central finite differences. The combination of these discretization methods is well known with wave equations (see, e.g., [10]). The methods related to spectral elements are studied in context of the time dependent acousto-elastic problem with second-order time-stepping schemes for instance in references [11,12,6].…”

Legendre spectral element method for solving sine-Gordon equation

“…[38]) Suppose that u ∈ C 2 (0, T; H s (Ω)) ∩ C 4 (0, T; L 2 (Ω)) is the exact solution of M u n+1 -2u n + u n-1 k Su n = 0 and U is the approximation result of the spectral element method under stability conditions on k. Then, for all t n > 0, we have u(t n ) -U n L 2 (Ω) ≤ O h min(N,s) N -s + k 2 .…”