Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations

In this paper, we develop a second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin (DG) scheme for the Kerr–Debye model. By using the approach first introduced by Zhang and Shu in [Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM J. Numer. Anal. 42 (2004) 641–666.] with an energy estimate and Taylor expansion, the asymptotic-preserving property of the semi-discrete DG methods is proved rigorously. In addition, we propose a class of unconditional positivity-preserving implicit–explicit (IMEX) Runge–Kutta methods for the system of ordinary differential equations arising from the semi-discretization of the Kerr–Debye model. The new IMEX Runge–Kutta methods are based on the modification of the strong-stability-preserving (SSP) implicit Runge–Kutta method and have second-order accuracy. The numerical results validate our analysis.

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Positivity-Preserving Well-Balanced Arbitrary Lagrangian–Eulerian Discontinuous Galerkin Methods for the Shallow Water Equations

Journal of Scientific Computing ◽

10.1007/s10915-021-01578-w ◽

2021 ◽

Vol 88 (3) ◽

Author(s):

Weijie Zhang ◽

Yinhua Xia ◽

Yan Xu

Keyword(s):

Shallow Water ◽

Discontinuous Galerkin ◽

Shallow Water Equations ◽

Discontinuous Galerkin Methods ◽

Galerkin Methods ◽

Arbitrary Lagrangian Eulerian ◽

Positivity Preserving

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An asymptotic preserving semi-implicit multiderivative solver

Applied Numerical Mathematics ◽

10.1016/j.apnum.2020.09.004 ◽

2021 ◽

Vol 160 ◽

pp. 84-101 ◽

Cited By ~ 1

Author(s):

Jochen Schütz ◽

David C. Seal

Keyword(s):

Asymptotic Preserving

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Positivity preserving high order schemes for angiogenesis models

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2021-0112 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

A. Carpio ◽

E. Cebrian

Keyword(s):

Kinetic Equation ◽

Time Discretization ◽

Strong Stability ◽

High Order ◽

Angiogenic Factor ◽

Strong Stability Preserving ◽

Positivity Preserving ◽

Blood Vessel Formation ◽

High Order Schemes ◽

Asymptotic Reduction

Abstract Hypoxy induced angiogenesis processes can be described by coupling an integrodifferential kinetic equation of Fokker–Planck type with a diffusion equation for the angiogenic factor. We propose high order positivity preserving schemes to approximate the marginal tip density by combining an asymptotic reduction with weighted essentially non oscillatory and strong stability preserving time discretization. We capture soliton-like solutions representing blood vessel formation and spread towards hypoxic regions.

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