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method of characteristics; finite differences; convection-diffusion problem; local error-estimate; stability
We describe a numerical method for the equation $u_t + pu_x - \varepsilon u_{xx} = f$ in $(0,1) \times (0,T)$ with Dirichlet boundary and initial conditions which is a combination of the method of characteristics and the finite-difference method. We prove both an a priori local error-estimate of a high order and stability. Example 3.3 indicates that our approximate solutions are disturbed only by a minimal amount of the artificial diffusion.
[1] M.B. Allen, A. Khosravani: Solute transport via alternating-direction collocation using the modified method of characteristics. Advances in Water Recources 15 (1992), 125–132. DOI 10.1016/0309-1708(92)90039-5
[2] I.S. Beresin, N.P. Shidkov: Numerical methods I. Nauka, Moscow, 1966. (Russian)
[3] J.H. Bramble, B.E. Hubbard: New monotone type approximations for elliptic problems. Math. Comp. (1964), no. 18, 349–367. DOI 10.1090/S0025-5718-1964-0165702-X | MR 0165702
[4] D.A. Bugai: Accuracy analysis of the eulerian-lagrangian numerical schemes for the convection-diffusion equation. Preprint.
[5] J. Dalík: A finite difference method for a two-dimensional convection-diffusion problem with dominating convection. Submitted to publication. MR 1463685
[6] J. Dougals Jr., T.F. Russell: Numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite elements or finite difference procedures. SIAM J. Numer. Anal. (1982), no. 19, 871–885. DOI 10.1137/0719063 | MR 0672564
[7] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uraltseva: Linear and quasilinear equations of parabolic type. Nauka, Moscow, 1967. (Russian)
[8] J.D. Lambert: Computational Methods in Ordinary Differential Equations. John Wiley & Sons, London, 1973. MR 0423815 | Zbl 0258.65069
[9] J.B. Noye: Finite-difference methods for solving the one-dimensional transport equation. Numerical modeling: Application to Marine Systems, J. Noye (ed.), Elsevier, North Holland, 1987, pp. 231–256. MR 0924023
[10] P.A. Raviart: Les méthodes d’élements finis en mécanique des fluides II. 3. Edditions Eyrolles, Paris, 1981. MR 0631851
[11] Y. Tourigny, E. Süli: The finite element method with nodes moving along the characteristics for convection-diffusion equations. Numer. Math. (1991), no. 59, 399–412. DOI 10.1007/BF01385788 | MR 1113198
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