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Elasto-acoustic wave propagation in geophysical media using hybrid high-order methods on general meshes

Romain Mottier, Alexandre Ern, Laurent Guillot

TL;DR

This work addresses accurate elasto-acoustic wave propagation in heterogeneous geophysical media by applying Hybrid High-Order (HHO) space discretization combined with Runge-Kutta time stepping. It reveals that efficient time integration can be achieved via static condensation—condensing face unknowns for explicit schemes and cell unknowns for implicit schemes—resulting in block-diagonal algebraic structures and enabling practical CFL analysis. The study provides numerical CFL estimates for explicit schemes, compares explicit and implicit time-stepping efficiency, and demonstrates the method on a realistic 2D basin-like configuration with general meshes, including hanging nodes, achieving accuracy comparable to tensor-product spectral element solvers. The results highlight HHO’s geometric flexibility, high-order accuracy, and suitability for complex geophysical geometries in coupled elasto-acoustic problems.

Abstract

Hybrid high-order (HHO) methods are numerical methods characterized by several interesting properties such as local conservativity, geometric flexibility and high-order accuracy. Here, HHO schemes are studied for the space semi-discretization of coupled elasto-acoustic waves in the time domain using a first-order formulation. Explicit and singly diagonal implicit Runge--Kutta (ERK & SDIRK) schemes are used for the time discretization. We show that an efficient implementation of explicit (resp. implicit) time schemes calls for a static condensation of the face (resp. cell) unknowns. Crucially, both static condensation procedures only involve block-diagonal matrices. Then, we provide numerical estimates for the CFL stability limit of ERK schemes and present a comparative study on the efficiency of explicit versus implicit schemes. Our findings indicate that implicit time schemes remain competitive in many situations. Finally, simulations in a 2D realistic geophysical configuration are performed, illustrating the geometrical flexibility of the HHO method: both hybrid (triangular and quadrilateral) and nonconforming (with hanging nodes) meshes are easily handled, delivering results of comparable accuracy to a reference spectral element software based on tensorized elements.

Elasto-acoustic wave propagation in geophysical media using hybrid high-order methods on general meshes

TL;DR

This work addresses accurate elasto-acoustic wave propagation in heterogeneous geophysical media by applying Hybrid High-Order (HHO) space discretization combined with Runge-Kutta time stepping. It reveals that efficient time integration can be achieved via static condensation—condensing face unknowns for explicit schemes and cell unknowns for implicit schemes—resulting in block-diagonal algebraic structures and enabling practical CFL analysis. The study provides numerical CFL estimates for explicit schemes, compares explicit and implicit time-stepping efficiency, and demonstrates the method on a realistic 2D basin-like configuration with general meshes, including hanging nodes, achieving accuracy comparable to tensor-product spectral element solvers. The results highlight HHO’s geometric flexibility, high-order accuracy, and suitability for complex geophysical geometries in coupled elasto-acoustic problems.

Abstract

Hybrid high-order (HHO) methods are numerical methods characterized by several interesting properties such as local conservativity, geometric flexibility and high-order accuracy. Here, HHO schemes are studied for the space semi-discretization of coupled elasto-acoustic waves in the time domain using a first-order formulation. Explicit and singly diagonal implicit Runge--Kutta (ERK & SDIRK) schemes are used for the time discretization. We show that an efficient implementation of explicit (resp. implicit) time schemes calls for a static condensation of the face (resp. cell) unknowns. Crucially, both static condensation procedures only involve block-diagonal matrices. Then, we provide numerical estimates for the CFL stability limit of ERK schemes and present a comparative study on the efficiency of explicit versus implicit schemes. Our findings indicate that implicit time schemes remain competitive in many situations. Finally, simulations in a 2D realistic geophysical configuration are performed, illustrating the geometrical flexibility of the HHO method: both hybrid (triangular and quadrilateral) and nonconforming (with hanging nodes) meshes are easily handled, delivering results of comparable accuracy to a reference spectral element software based on tensorized elements.

Paper Structure

This paper contains 9 sections, 10 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model problem
  3. Strong formulation
  4. Weak formulation
  5. HHO space semi-discretization
  6. Mesh
  7. Degrees of freedom
  8. Space semi-discrete formulation and algebraic realization
  9. Degrees of freedom.

Figures (3)

  • Figure 1: Solid subdomain $\Omega^{\sc{s}}$ composed of the sedimentary basin and the bedrock, fluid subdomain $\Omega^{\sc{f}}$, and unit normal $\boldsymbol{n}_\Gamma$ along the interface $\Gamma$ (highlighted with a thick black line)
  • Figure 2: Generic example of a mesh with hanging nodes at the interface $\Gamma$. Filled elements are reinterpreted: filled squares as hexagons and filled triangles as quadrilaterals. Unfilled elements are treated for what they are: squares and triangles.
  • Figure 3: Elasto-acoustic dofs along a solid-fluid interface, with hexagonal mesh cells and lowest equal-order discretization $(k^\prime = k = 1)$.