Efficient upwind and partitioned implicit/explicit finite difference schemes for the second-order wave equation on overset grids
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Authors
Carson, Allison
Issue Date
2024-05
Type
Electronic thesis
Thesis
Thesis
Language
en_US
Keywords
Applied mathematics
Alternative Title
Abstract
In this thesis, we discuss multiple finite-difference methods to solve the wave equation in second-order form on complex geometry discretized by overset grids. We begin by incorporating upwind dissipation into the modified equation (ME) scheme in an efficient and accurate manner in order to suppress instabilities that can grow from small perturbations in the computation, as the ME scheme does not have strong dissipative effects. For stiff problems, we derive an implicit reformulation of the ME scheme to give unconditional stability. A hybrid implicit-explicit ME scheme is also devised for such problems, especially for those with domains containing localized areas of finer grid resolution that can be caused by intricate geometry. From a numerical perspective, one significant challenge for discretizations of the wave equation is their lack of inherent dissipation mechanism. As a result, perturbations, such as those arising from overset grids, can lead to numerical instability. These instabilities can be eliminated through the addition of artificial dissipation devices, and for wave equations in second-order form, there are number of existing formulations. Current methods include ad-hoc dissipation (FDA), which adds an artificial dissipative operator into the ME scheme, and the more recently developed upwind schemes (UW). Both options have been shown to stabilize grid solutions, but existing FDA methods may loose an order of accuracy when strong instabilities are present and existing UW formulations are computationally expensive and complicated in implementation. Here we devise the upwind predictor-corrector (UPC) scheme at general even-order to introduce upwind dissipation into the ME scheme in a simple, accurate, and efficient manner. In this approach a higher-order upwind operator, derived from the modified equation analysis of the UW scheme, is incorporated into the ME scheme through a modular corrector-style stage. The higher-order operator maintains the underlying ME scheme’s accuracy as well as its time-step stability restriction, and takes on the dissipative properties of the UW scheme, while avoiding much of the UW scheme’s high cost and complex implementation. The scheme is extended to curvilinear coordinates, and numerical tests illustrate the UPC scheme’s accuracy and stability on single and overset grids in one, two, and three dimensions. Various CPU run time comparisons on rectangular and curvilinear grids demonstrate the UPC scheme’s computational speedups over the UW schemes. Explicit schemes, however, can be inefficient for stiff problems due to their time-step restrictions. Such stiffness may arise in problems with small grid cells associated with meshing for complex geometry or in algorithms requiring long time integrations. To address these situations, we derive the implicit modified equation (IME) schemes at second- and fourth-order accuracy. These schemes adopt the compact stencil of the ME scheme and introduce implicitness by applying spatial operators to a time-average approximation spanning three levels in time. Truncation error analysis reveals a parametric class of schemes, and stability analysis identifies those with unconditional stability. The upwind operator derived from the UW scheme is also incorporated into the IME schemes to ensure stability on overset grids. Three different methods to incorporate upwinding are investigated, and stability analysis reveals conditions for stability. Numerical results confirm the accuracy and stability of all the methods. Results on overlapping and curvilinear grids in one, two, and three dimensions demonstrate the stability of the dissipative schemes. For the case of stiffness induced by locally refined grids, we devise the partitioned implicit-explicit (PIE) scheme. In this approach we use local implicit time stepping in areas of the domain exhibiting stiffness, and explicit time integration elsewhere. The time step is then selected based on the explicit scheme, which in this case is the ME scheme. Numerical analyses and example calculations demonstrate the stability of the method. The PIE scheme is also made dissipative by replacing the IME and ME schemes with their upwind counterparts, the UPC scheme and the upwind IME schemes. Numerical tests on overset grids in one, two, and three multiple dimensions confirm the accuracy and stability of the schemes. Simulations on complex geometry show the PIE scheme’s superiority in computational efficiency over the UPC scheme in stiff problems.
Description
May2024
School of Science
School of Science
Full Citation
Publisher
Rensselaer Polytechnic Institute, Troy, NY