A hybrid numerical method for elastic wave propagation in discontinuous media with complex geometry

TL;DR

This paper tackles stable, high-order simulation of elastic waves in media with discontinuities and complex geometry by introducing a hybrid FD-DG method that couples SBP finite differences on Cartesian grids with IPDG on unstructured meshes through norm-compatible projection operators. The main contributions are the construction and optimization of the interface projections to achieve discrete energy conservation and optimal convergence in the second-order displacement formulation, along with a rigorous energy-based stability analysis. Numerical results across continuous, discontinuous, curved, and geometrically complex domains demonstrate energy conservation, robust convergence at rates matching theory, and improved conditioning when projection operators are spectrum-optimized. The approach enables accurate, efficient seismic-scale simulations in heterogeneous media with complex interfaces and geometries.

Abstract

We develop a high order accurate numerical method for solving the elastic wave equation in second-order form. We hybridize the computationally efficient Cartesian grid formulation of finite differences with geometrically flexible discontinuous Galerkin methods on unstructured grids by a penalty based technique. At the interface between the two methods, we construct projection operators for the pointwise finite difference solutions and discontinuous Galerkin solutions based on piecewise polynomials. In addition, we optimize the projection operators for both accuracy and spectrum. We prove that the overall discretization conserves a discrete energy, and verify optimal convergence in numerical experiments.

Figures (8)

• Figure 1: An illustration of the FD mesh $\mathbf{x}_u$, the intermediary piecewise continuous grid $\mathscr{T}_{\Tilde{u}}$, glue grid $\mathscr{T}_g$ and DG grid $\mathscr{T}_v$. Although the piecewise continuous intervals are drawn as if the nodes are overlapping, each element should be thought of being disjoint with repeated edge nodes.
• Figure 2: (a) An illustration of the grid on the interface (b) The largest absolute eigenvalues $\max_i|e_i|$ of the discretization matrix rescaled by $h^2$ as a function of $\delta$. Note the small interval along the $y$-axis.
• Figure 3: (a) The coarsest discretization of the domain used with matching grids (b) The discrete error in $||\cdot||_h$-norm at the final time $T=1$.
• Figure 4: (a) An illustration of the mesh structure used for mathing degrees of freedom with $p=3$, corresponding to $N=31$. (b) Numerical errors in $\|\cdot\|_h$ norm of the fourth and sixth order experiments for matching degrees of freedom along the interface.
• Figure 5: (a) A coarse discretization of the domain demonstrating the linear approximation of the southern boundary given by $f_s(x)$ (b) The discrete error at final time $T=1$ in $||\cdot||_h$-norm when the material is continuous and the bottom boundary of the DG domain is curvilinear, given by $f_s(x)=0.4x(x-1)e^{-10(x-0.5)^2}$ for both the fourth- and sixth-order methods.

Theorems & Definitions (11)

• Definition 2.1
• Definition 2.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Definition 3.1
• Corollary 3.1
• Theorem 3.1
• proof