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

Kenny Wiratama, Kenneth Duru, Yunho Kim

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.

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

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.
Paper Structure (22 sections, 17 theorems, 66 equations, 12 figures, 2 tables, 1 algorithm)

This paper contains 22 sections, 17 theorems, 66 equations, 12 figures, 2 tables, 1 algorithm.

Table of Contents

  1. The continuous problem
  2. Forward problem
  3. Adjoint problem
  4. Numerical discretization
  5. The semi-discrete problem
  6. Forward problem
  7. Adjoint problem
  8. The fully discrete problem
  9. Forward problem
  10. Adjoint problem
  11. Numerical experiments
  12. One-dimensional convergence test
  13. One-dimensional Gaussian inverse problem
  14. Two-dimensional convergence test
  15. Two-dimensional Gaussian inverse problem
  16. ...and 7 more sections

Key Result

Theorem 1

Consider the IBVP contwaveeq, initcond, wavebcs with $\sigma^2 \ge 0$ and homogeneous boundary data $b_L = b_R = 0$. If the boundary parameters $\alpha_L, \beta_L, \alpha_R, \beta_R$ satisfy the conditions constantscond, then in a source-free medium $(S = 0)$, the energy contenergy satisfies the equ where $\operatorname{BT}(t) = \operatorname{BT}_L(t) + \operatorname{BT}_R(t)$.

Figures (12)

  • Figure 1: Log-scale error plots for the undamped wave equation $\sigma = 0$ using the DP-SBP operators of orders (a) 2, (b) 4, (c) 6 and (d) 8 with different second time derivative approximations.
  • Figure 2: Log-scale error plots for the damped wave equation $\sigma = 1$ using the DP-SBP operators of orders (a) 2, (b) 4, (c) 6 and (d) 8 with different second time derivative approximations.
  • Figure 3: The initial guess and the exact initial displacement profile to be recovered from boundary measurements in the 1D numerical inverse simulations.
  • Figure 4: Iterative numerical inversion to recover the 1D Gaussian initial displacement field for the undamped wave equation $\sigma = 0$: (a) iteration 1, (b) iteration 5, (c) error vs. iteration.
  • Figure 5: Iterative numerical inversion to recover the 1D Gaussian initial displacement field for the damped wave equation $\sigma = 1$: (a) iteration 1, (b) iteration 5, (c) error vs. iteration.
  • ...and 7 more figures

Theorems & Definitions (27)

  • Theorem 1
  • proof
  • Corollary 1
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4
  • Remark 1
  • Theorem 5
  • proof
  • ...and 17 more