# A Predictor-Corrector Scheme for Conservation Equations with Discontinuous Coefficients

## Authors

• Nasrin Okhovati Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, 7635131167
• Mohammad Izadi Department of Applied Mathematics, Shahid Bahonar University of Kerman, Kerman, 7616913439

## Keywords:

discontinuous flux, finite difference approximation, MacCormack scheme

## Abstract

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.

## Author Biographies

### Nasrin Okhovati, Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, 7635131167

Department of Mathematics

### Mohammad Izadi, Department of Applied Mathematics, Shahid Bahonar University of Kerman, Kerman, 7616913439

Departmanet of Appiled Mathematics

## References

Gimse, T. & Risebro, N.H., Solution of the Cauchy Problem for a Conservation Law with a Discontinuous Flux Function, SIAM J. Math. Anal., 23(3), pp. 635-648, 1992.

Diehl, S., A Conservation Law with Point Source and Discontinuous Flux Function Modelling Continuous Sedimentation, SIAM J. Appl. Math., 56(2), pp. 388-419, 1996.

B1/4rger, R., Diehl, S. & Mart, M.D.C, A System of Conservation Laws with Discontinuous Flux Modelling Flotation with Sedimentation, IMA J. Appl. Math., 84, pp. 930-973, 2019.

B1/4rger, R., Karlsen, K.H., Risebro, N.H. & Towers, J.D., Numerical Methods for the Simulation of Continuous Sedimentation in Ideal Clarifier-Thickener Units, Int. J. Miner. Process., 73(2-4), pp. 209-228, 2004.

Mochon, S., An Analysis of the Traffic on Highways with Changing Surface Conditions, Math. Modelling, 9, pp. 1-11, 1987.

Holden, H. & Risebro, N.H., A Mathematical Model of Traffic Flow on a Network of Unidirectional Roads, SIAM J. Math. Anal., 26(4), pp. 999-1017, 1995.

Andreianov, B., Karlsen, K.H. & Risebro, N.H., A Theory of L1-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux, Arch. Rational Mech. Anal., 201, pp. 27-86, 2011.

Izadi, M., Streamline Diffusion Methods for Treating the Coupling Equations of two Hyperbolic Conservation Laws, Math. Comput. Model., 45, pp. 201-214, 2007.

Izadi, M., A Posteriori Error Estimates for the Coupling Equations of Scalar Conservation Laws, BIT. Numer. Math., 49(4), pp. 697-720, 2009.

Izadi, M., Application of the Newton-Raphson Method in a SDFEM for Inviscid Burgers Equation, Comput. Methods Differ. Equ., 8(4), pp. 708-732, 2020.

Karlsen, K.H., Risebro, N.H. & Towers, J.D., Upwind Difference Approximations for Degenerate Parabolic Convection-Diffusion Equations with a Discontinuous Coefficient, IMA J. Numer. Anal., 22(4), pp. 623-664, 2002

Wen, X., Jin, S., Convergence of an Immersed Interface Upwind Scheme for Linear Advection Equations with Piecewise Constant Coefficients I: L1-Error Estimates, J. Comput. Math., 26, pp. 1-22, 2008.

Okhovati, N. & Izadi, M., Numerical Coupling of Two Scalar Conservation Laws by a RKDG Method, J. Korean Soc. Ind. Appl. Math., 23(3), pp. 211-236, 2019.

Mishra, S., Numerical Methods for Conservation Laws with Discontinuous Coefficients, In Handbook of Numerical Analysis, 18, pp. 479-506. Elsevier, 2017.

LeVeque, R.J., Numerical Methods for Conservation Laws, Birkhauser-Verlag, 1992.

KruA3/4kov, S.N., First Order Quasilinear Equationsin Several Independent Variables, USSR Math. Sbornik, 10(2), pp. 217-243, 1970.

Karlsen, K.H., Risebro, N.H. & Towers, J.D., On a Nonlinear Degenerate Parabolic Transport-Diffusion Equation with a Discontinuous Coefficient, Electron. J. Diff. Eqs., 23, pp. 1-23, 2002.

MacCormack, R. W., The Effect of Viscosity in Hypervelocity Impact Cratering, AIAA Paper, pp. 69-354, 1969.

Izadi, M. (in press), Two-Stages Explicit Schemes Based Numerical Approximations of Convection-Diffusion Equations, Int. J. Comput. Sci. Math., 2020.

Kulmart, K. & Pochai, N., Numerical Simulation for Salinity Intrusion Measurement Models Using the MacCormack Finite Difference Method with Lagrange Interpolation, J. Interdiscip. Math., 23(6), pp. 1157-1185, 2020.

Hoffmann, K.A. & Chiang, S.T., Computational Fluid Dynamics for Engineers, 4th Ed., Vols. I & II, Engineering Education System, 2000.