Core Concepts
The author presents a new efficient discretization method for the Laplacian operator on complex geometries by extending the continuous summation-by-parts (SBP) framework to second derivatives and combining it with spectral-type SBP operators on Gauss-Lobatto quadrature points.
Abstract
The author addresses the challenge of efficiently discretizing the Laplacian operator on complex geometries, which is of primary interest in academia and industry for problems involving second derivatives, such as the Navier-Stokes equations or wave propagation problems.
The key highlights and insights are:
The author extends the continuous summation-by-parts (SBP) framework to second derivatives and combines it with spectral-type SBP operators on Gauss-Lobatto quadrature points to obtain a highly efficient discretization of the Laplacian on complex domains.
The resulting Laplace operator is defined on a grid without duplicated points on the interfaces, removing unnecessary degrees of freedom in the scheme, and is proven to satisfy a discrete equivalent to Green's first identity.
The author proves semi-discrete stability using the new Laplace operator for the acoustic wave equation in 2D.
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 conducted on the acoustic wave equation in 2D, demonstrating the accuracy and efficiency properties of the method, as well as its potential use in a realistic problem with a complex region of the domain discretized using the new method and coupled to the rest of the domain discretized using a traditional finite difference method.
Stats
The author presents the following key figures and metrics:
"The error results with the SBP GL operators are also presented in Table 1, including convergence rates estimated as q = log(e1/e2)/log(N2/N1)^(1/2), where e1 and e2 are the L2-errors with N1 and N2 DOFs."