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Efficient discretization of the Laplacian on complex geometries

Gustav Eriksson

Abstract

Highly accurate simulations of problems including second derivatives on complex geometries are of primary interest in academia and industry. Consider for example the Navier-Stokes equations or wave propagation problems of acoustic or elastic waves. Current finite difference discretization methods are accurate and efficient on modern hardware, but they lack flexibility when it comes to complex geometries. In this work I extend the continuous summation-by-parts (SBP) framework to second derivatives and combine it with spectral-type SBP operators on Gauss-Lobatto quadrature points to obtain a highly efficient discretization (accurate with respect to runtime) of the Laplacian on complex domains. The resulting Laplace operator is defined on a grid without duplicated points on the interfaces, thus removing unnecessary degrees of freedom in the scheme, and is proven to satisfy a discrete equivalent to Green's first identity. Semi-discrete stability using the new Laplace operator is proven for the acoustic wave equation in 2D. Furthermore, the method can easily be coupled together with traditional finite difference operators using glue-grid interpolation operators, resulting in a method with great practical potential. Two numerical experiments are done on the acoustic wave equation in 2D. First on a problem with an analytical solution, demonstrating the accuracy and efficiency properties of the method. Finally, a more realistic problem is solved, where a complex region of the domain is discretized using the new method and coupled to the rest of the domain discretized using a traditional finite difference method.

Efficient discretization of the Laplacian on complex geometries

Abstract

Highly accurate simulations of problems including second derivatives on complex geometries are of primary interest in academia and industry. Consider for example the Navier-Stokes equations or wave propagation problems of acoustic or elastic waves. Current finite difference discretization methods are accurate and efficient on modern hardware, but they lack flexibility when it comes to complex geometries. In this work I extend the continuous summation-by-parts (SBP) framework to second derivatives and combine it with spectral-type SBP operators on Gauss-Lobatto quadrature points to obtain a highly efficient discretization (accurate with respect to runtime) of the Laplacian on complex domains. The resulting Laplace operator is defined on a grid without duplicated points on the interfaces, thus removing unnecessary degrees of freedom in the scheme, and is proven to satisfy a discrete equivalent to Green's first identity. Semi-discrete stability using the new Laplace operator is proven for the acoustic wave equation in 2D. Furthermore, the method can easily be coupled together with traditional finite difference operators using glue-grid interpolation operators, resulting in a method with great practical potential. Two numerical experiments are done on the acoustic wave equation in 2D. First on a problem with an analytical solution, demonstrating the accuracy and efficiency properties of the method. Finally, a more realistic problem is solved, where a complex region of the domain is discretized using the new method and coupled to the rest of the domain discretized using a traditional finite difference method.
Paper Structure (10 sections, 2 theorems, 50 equations, 5 figures, 1 table)

This paper contains 10 sections, 2 theorems, 50 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. One-dimensional SBP operators
  4. Two-dimensional operators
  5. The continuous SBP Laplace operator
  6. The acoustic wave equation
  7. Numerical results
  8. Accuracy
  9. Acoustic simulation
  10. Conclusions

Key Result

Lemma 3.1

The reduced Laplace operator where satisfies the SBP property where the boundary terms are given by

Figures (5)

  • Figure 1: Two-block curvilinear example domain. The boundaries are color-coded as follows: the interface $\partial \Omega_I$ is light green, the north boundaries $\partial \Omega_1$ are dark green, the south boundaries $\partial \Omega_2$ are dark blue, and the side boundaries $\partial \Omega_3$ are light blue.
  • Figure 2: Domain decompositions of the unit circle.
  • Figure 3: $L_2$-error versus degrees of freedom and runtime for the continuous SBP-SAT method with SBP GL operators, the SBP-P-SAT method with traditional SBP operators, and the SBP-P-SAT method with boundary optimized operators.
  • Figure 4: Boundaries (left figure) and block decomposition (right figure) of domain with complex shapes. The boundaries are color-coded in the left figure as follows: the interface $\partial \Omega_I$ is light green, the Dirichlet boundary $\partial \Omega_1$ is dark green, the Neumann boundaries $\partial \Omega_2$ are dark blue, and the outflow boundaries $\partial \Omega_3$ are light blue.
  • Figure 5: Numerical simulation results at $t = 50$ ms, $t = 75$ ms, $t = 100$ ms, and $t = 200$ ms of acoustic pressure waves emitting from a source in the upper block (water) and propagating into the lower block (soil) and reflecting off of the complex objects.

Theorems & Definitions (7)

  • Definition 1
  • Remark 1
  • Remark 2
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof