scholarly journals An Engquist–Osher type finite difference scheme with a discontinuous flux function in space

2011 ◽  
Vol 235 (17) ◽  
pp. 4966-4977 ◽  
Author(s):  
Guodong Wang
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Flux Function ◽  
Discontinuous Flux Function ◽  
Discontinuous Flux

scholarly journals A Predictor-Corrector Scheme for Conservation Equations with Discontinuous Coefficients

2020 ◽  
Vol 52 (3) ◽  
pp. 322-338
Author(s):  
Nasrin Okhovati ◽  
Mohammad Izadi
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Discontinuous Coefficients ◽  
Flow In Porous Media ◽  
Two Phase ◽  
Flux Function ◽  
Uniform Grids ◽  
Discontinuous Flux ◽  
Predictor Corrector

In this paper we propose an explicit predictor-corrector finite difference scheme to numerically solve one-dimensional conservation laws with discontinuous flux function appearing in various physical model problems, such as traffic flow and two-phase flow in porous media. The proposed method is based on the second-order MacCormack finite difference scheme and the solution is obtained by correcting first-order schemes. It is shown that the order of convergence is quadratic in the grid spacing for uniform grids when applied to problems with discontinuity. To illustrate some properties of the proposed scheme, numerical results applied to linear as well as non-linear problems are presented.

scholarly journals The convergence of a numerical method for total variation flow

2021 ◽  
Vol 15 ◽  
pp. 174830262110113
Author(s):  
Qianying Hong ◽  
Ming-jun Lai ◽  
Jingyue Wang
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Total Variation ◽  
Iterative Algorithm ◽  
Finite Difference Scheme ◽  
Gradient Flow ◽  
Time Dependent ◽  
Total Variation Flow

We present a convergence analysis for a finite difference scheme for the time dependent partial different equation called gradient flow associated with the Rudin-Osher-Fetami model. We devise an iterative algorithm to compute the solution of the finite difference scheme and prove the convergence of the iterative algorithm. Finally computational experiments are shown to demonstrate the convergence of the finite difference scheme.

scholarly journals A Laplace transform finite difference scheme for the Fisher-KPP equation

2021 ◽  
Vol 15 ◽  
pp. 174830262199958
Author(s):  
Colin L Defreitas ◽  
Steve J Kane
Keyword(s):  
Finite Difference ◽  
Laplace Transform ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Finite Difference Methods ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Travelling Wave ◽  
Reaction Diffusion Equation ◽  
Kpp Equation

This paper proposes a numerical approach to the solution of the Fisher-KPP reaction-diffusion equation in which the space variable is developed using a purely finite difference scheme and the time development is obtained using a hybrid Laplace Transform Finite Difference Method (LTFDM). The travelling wave solutions usually associated with the Fisher-KPP equation are, in general, not deemed suitable for treatment using Fourier or Laplace transform numerical methods. However, we were able to obtain accurate results when some degree of time discretisation is inbuilt into the process. While this means that the advantage of using the Laplace transform to obtain solutions for any time t is not fully exploited, the method does allow for considerably larger time steps than is otherwise possible for finite-difference methods.

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