A space-time dual-pairing summation-by-parts framework for forward and adjoint wave equations

TL;DR

This paper introduces a novel space-time dual-pairing summation-by-parts framework for wave equations, achieving high-order accuracy, stability, and adjoint consistency for forward and inverse problems.

Contribution

It presents the first space-time DP-SBP framework that combines high-order accuracy, stability, and adjoint consistency for wave propagation simulations.

Findings

Achieves high-order accuracy in space and time.

Ensures stability through energy estimates.

Demonstrates convergence in numerical experiments.

Abstract

In this paper, we propose the first of its kind space-time dual-pairing summation by parts (DP-SBP) numerical framework for forward and adjoint wave propagation problems. This novel approach enables us to achieve spatial and temporal high order accuracy while naturally introducing dissipation in time. Within this framework, initial and boundary conditions are weakly imposed using the simultaneous approximation term (SAT) technique. Fully discrete energy estimates are derived, ensuring the stability of the resulting numerical scheme. Furthermore, the proposed space-time numerical framework allows us to construct adjoint consistent fully discrete numerical approximations, which can be applied to solve inverse wave propagation problems. We provide numerical experiments in one and two spatial dimensions to verify the theoretical analysis and demonstrate convergence of numerical errors.