A time-continuous embedding method for scalar hyperbolic conservation laws on manifolds
type: Item
instance of: scholarly article
title: A time-continuous embedding method for scalar hyperbolic conservation laws on manifolds
zbMATH Open document ID: 1504.65187
author: Yinghua Wang
author: Baoshan Wang
author: L. Ling
author: Wai Sun Don
published in: Journal of Scientific Computing
publication date: 2022-12-13T00:00:00Z
review text: A new closest point TC (Time-Continuous) embedding method is developed for solving nonlinear scalar hyperbolic conservation law (PDEs) on one-dimensional, connected, smooth, and closed manifolds (SPDEs). It improves upon the classical CP (Closest Point) embedding method that requires a costly re-establishment of the extension function's CAN (Constant-Along-Normal) property at every time step; the TC embedding method incorporates the CAN-property analytically and explicitly in the SPDEs directly. The TC embedding SPDEs are solved by the second-order nonlinear central finite volume scheme with a nonlinear minmod slope limiter in space, and the third-order total variation diminished Runge-Kutta scheme in time. An adaptive nonlinear essentially non-oscillatory polynomial interpolation is used to obtain the solution values at the ghost cells. Numerical results in solving the linear wave equation and the Burgers' equation show that the proposed TC embedding method has better accuracy, improved resolution, and reduced CPU times than the classical CP embedding method. The Burgers' equation, the traffic flow problem, and the Buckley-Leverett equation are solved to demonstrate the efficiency of the TC embedding method in resolving fine-scale structures efficiently even in the presence of a shock and the essentially non-oscillatory capturing of shocks and rarefaction waves on simple and complex shaped one-dimensional manifolds. Burgers' equation is also solved on the two-dimensional torus-shaped and spherical-shaped manifolds.
reviewed by: Abdallah Bradji
Mathematics Subject Classification ID: 65M08
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65N08
Mathematics Subject Classification ID: 65D05
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 76P05
Mathematics Subject Classification ID: 76M12
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35Q53
zbMATH DE Number: 7632683
zbMATH Keywords: scalar hyperbolic conservation laws
zbMATH Keywords: time-continuous embedding
zbMATH Keywords: closest point embedding
zbMATH Keywords: central finite volume
zbMATH Keywords: ENO interpolation
zbMATH Keywords: Burgers'
zbMATH Keywords: traffic flow
zbMATH Keywords: Buckley-Leverett
MaRDI profile type: Publication
full work available at URL: s10915-022-02023-2
OpenAlex ID: W4308749388
Wikidata QID: Q115603741
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DOI: 10.1007/S10915-022-02023-2
description: scientific article; zbMATH DE number 7632683