{"title":"An Efficient Backward Semi-Lagrangian Scheme for Nonlinear Advection-Diffusion Equation","authors":"Soyoon Bak, Sunyoung Bu, Philsu Kim","volume":92,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":1104,"pagesEnd":1108,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/9999046","abstract":"
In this paper, a backward semi-Lagrangian scheme
\r\ncombined with the second-order backward difference formula
\r\nis designed to calculate the numerical solutions of nonlinear
\r\nadvection-diffusion equations. The primary aims of this paper are
\r\nto remove any iteration process and to get an efficient algorithm
\r\nwith the convergence order of accuracy 2 in time. In order to achieve
\r\nthese objects, we use the second-order central finite difference and the
\r\nB-spline approximations of degree 2 and 3 in order to approximate
\r\nthe diffusion term and the spatial discretization, respectively. For the
\r\ntemporal discretization, the second order backward difference formula
\r\nis applied. To calculate the numerical solution of the starting point
\r\nof the characteristic curves, we use the error correction methodology
\r\ndeveloped by the authors recently. The proposed algorithm turns out
\r\nto be completely iteration free, which resolves the main weakness
\r\nof the conventional backward semi-Lagrangian method. Also, the
\r\nadaptability of the proposed method is indicated by numerical
\r\nsimulations for Burgers’ equations. Throughout these numerical
\r\nsimulations, it is shown that the numerical results is in good
\r\nagreement with the analytic solution and the present scheme offer
\r\nbetter accuracy in comparison with other existing numerical schemes.<\/p>\r\n","references":"[1] C. DEBOOR, A practical Guide to splines, Springer Verlag, New York,\r\n(1978).\r\n[2] G. H\u00a8aMMERLIN AND K. H. HOFFMANN, Numerical Mathematics,\r\nSpringer Verlag, New York, (1991).\r\n[3] P. KIM, X. PIAO AND S. D. KIM, An error corrected Eluer method for\r\nsolving stiff problems based on chebyshev collocation, SIAM J. Numer.\r\nAnal. 49 (2011) pp. 2211\u20132230.\r\n[4] S. D. KIM, X. PIAO AND P. KIM, Convergence on error correction\r\nmethods for solving initial value problems, J. Comput. Appl. Math., 236\r\n(2012) pp. 4448\u20134461.\r\n[5] J. L. MCGREGOR, Economical determination of departure points for\r\nsemi-Lagrangian models, Mon. Weather Rev., 121 (1993) pp. 221\u2013230 .\r\n[6] J. WANG AND A. LAYTON, New numerical methods for Burgers\u2019\r\nequation based on semi-Lagrangian and modified equation approaches,\r\nAppl. Numer. Math., 60 (2010) pp. 645\u2013657.\r\n[7] D. XIU AND G. E. KARNIADAKIS, A Semi-Lagrangian high-order\r\nmethod for Navier-Stokes equations J. Comput. Physics, 172 (2001) pp.\r\n658\u2013684.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 92, 2014"}