arXiv subjects

Christina G. Taylor

Publications and source records attributed to Christina G. Taylor.

4 recordsLinked to original sources

arXiv2026

Bessel Functions and Analysis of Circular Waveguides

The paper is devoted to the study of circularly coiled optical slab waveguides, which is also applicable to acoustical waveguides. We use a change of variables and the classical Frobenius method to compute Bessel functions of complex order and complex argument, and combine it with a perfectly matched layer technique to solve the relevant Bessel eigenvalue problem and deliver accurate loss factors for eigensolutions to the three-layer optical slab waveguide problem. The solutions provide a benchmark for verifying model implementations of this problem and allow for a numerical verification of the Glazman criterion that provides a foundation for the well-posedness and stability analysis of homogeneous circular waveguides with impedance boundary conditions.

physics.comp-ph↗

arXiv2024

An Entropy Stable High-Order Discontinuous Galerkin Method on Cut Meshes

High-order entropy stable summation-by-parts (SBP) schemes are a class of robust and accurate numerical methods for hyperbolic conservation laws that are numerically stable at arbitrary order without the need for artificial stabilization. While SBP schemes are well-established on simplicial and tensor-product elements, they have not been extended to cut meshes. Cut meshes provide a convenient and efficient means of mesh generation for domains with embedded boundaries but can be difficult to use due to their arbitrarily shaped cut elements. Using the skew-hybridized SBP formulation of Chan ["Skew-symmetric entropy stable...", JSC, 2019], we present a high-order accurate, entropy stable scheme for hyperbolic conservation laws on cut meshes. The formulation requires positive/non-negative weight quadrature rules on cut elements, which we construct via explicit parameterizations, subtriangulations, and Caratheodory pruning. We numerically verify the accuracy and stability of our method using the shallow water and compressible Euler equations and note promising results for the use of state redistribution with entropy stable methods.

math.NA↗

arXiv2024

An Energy Stable High-Order Cut Cell Discontinuous Galerkin Method with State Redistribution for Wave Propagation

Cut meshes are a type of mesh that is formed by allowing embedded boundaries to "cut" a simple underlying mesh resulting in a hybrid mesh of cut and standard elements. While cut meshes can allow complex boundaries to be represented well regardless of the mesh resolution, their arbitrarily shaped and sized cut elements can present issues such as the small cell problem, where small cut elements can result in a severely restricted CFL condition. State redistribution, a technique developed by Berger and Giuliani [1], can be used to address the small cell problem. In this work, we pair state redistribution with a high-order discontinuous Galerkin scheme that is $L_2$ energy stable for arbitrary quadrature. We prove that state redistribution can be added to a provably $L_2$ energy stable discontinuous Galerkin method on a cut mesh without damaging the scheme's $L_2$ stability. We numerically verify the high order accuracy and stability of our scheme on two-dimensional wave propagation problems.

math.NA↗

arXiv2020

Efficient computation of Jacobian matrices for entropy stable summation-by-parts schemes

Entropy stable schemes replicate an entropy inequality at the semi-discrete level. These schemes rely on an algebraic summation-by-parts (SBP) structure and a technique referred to as flux differencing. We provide simple and efficient formulas for Jacobian matrices for the semi-discrete systems of ODEs produced by entropy stable discretizations. These formulas are derived based on the structure of flux differencing and derivatives of flux functions, which can be computed using automatic differentiation (AD). Numerical results demonstrate the efficiency and utility of these Jacobian formulas, which are then used in the context of two-derivative explicit time-stepping schemes and implicit time-stepping.

math.NA↗