Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30379
An Efficient Backward Semi-Lagrangian Scheme for Nonlinear Advection-Diffusion Equation

Authors: Soyoon Bak, Sunyoung Bu, Philsu Kim


In this paper, a backward semi-Lagrangian scheme combined with the second-order backward difference formula is designed to calculate the numerical solutions of nonlinear advection-diffusion equations. The primary aims of this paper are to remove any iteration process and to get an efficient algorithm with the convergence order of accuracy 2 in time. In order to achieve these objects, we use the second-order central finite difference and the B-spline approximations of degree 2 and 3 in order to approximate the diffusion term and the spatial discretization, respectively. For the temporal discretization, the second order backward difference formula is applied. To calculate the numerical solution of the starting point of the characteristic curves, we use the error correction methodology developed by the authors recently. The proposed algorithm turns out to be completely iteration free, which resolves the main weakness of the conventional backward semi-Lagrangian method. Also, the adaptability of the proposed method is indicated by numerical simulations for Burgers’ equations. Throughout these numerical simulations, it is shown that the numerical results is in good agreement with the analytic solution and the present scheme offer better accuracy in comparison with other existing numerical schemes.

Keywords: Semi-Lagrangian method, iteration free method, nonlinear advection-diffusion equation

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 2021


[1] C. DEBOOR, A practical Guide to splines, Springer Verlag, New York, (1978).
[2] G. H¨aMMERLIN AND K. H. HOFFMANN, Numerical Mathematics, Springer Verlag, New York, (1991).
[3] P. KIM, X. PIAO AND S. D. KIM, An error corrected Eluer method for solving stiff problems based on chebyshev collocation, SIAM J. Numer. Anal. 49 (2011) pp. 2211–2230.
[4] S. D. KIM, X. PIAO AND P. KIM, Convergence on error correction methods for solving initial value problems, J. Comput. Appl. Math., 236 (2012) pp. 4448–4461.
[5] J. L. MCGREGOR, Economical determination of departure points for semi-Lagrangian models, Mon. Weather Rev., 121 (1993) pp. 221–230 .
[6] J. WANG AND A. LAYTON, New numerical methods for Burgers’ equation based on semi-Lagrangian and modified equation approaches, Appl. Numer. Math., 60 (2010) pp. 645–657.
[7] D. XIU AND G. E. KARNIADAKIS, A Semi-Lagrangian high-order method for Navier-Stokes equations J. Comput. Physics, 172 (2001) pp. 658–684.