A Survey of the Implementation of Numerical Schemes for Linear Advection Equation

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The interpolation method in a semi-Lagrangian scheme is decisive to its performance. Given the number of grid points one is considering to use for the interpolation, it does not necessarily follow that maximum formal accuracy should give the best results. For the advection equation, the driving force of this method is the method of the characteristics, which accounts for the flow of information in the model equation. This leads naturally to an interpolation problem since the foot point is not in general located on a grid point. We use another interpolation scheme that will allow achieving the high order for the box initial condition.

References

[1] Strikwerda, J.C. (1989) Finite Difference Schemes and Partial Differential Equations. Wadsworth & Brooks, USA.

[2] McRea, G.J. and Godin, W.R. (1967) Numerical Solution of Atmospheric Diffusion for Chemically Reacting Flows. Journal of Computational Physics, 77, 1-42.

[3] Morton, K.W. (1980) Stability of Finite Difference Approximations to a Diffusion-Convection Equation. International Journal for Numerical Methods in Engineering, 15, 677-683.

http://dx.doi.org/10.1002/nme.1620150505

[4] Hundsdorfer, W. and Koren, B. (1995) A Positive Finite-Difference Advection Scheme Applied on Locally Refined Grids. Journal of Computational Physics, 117, 35-36. http://dx.doi.org/10.1006/jcph.1995.1042

[5] Canuto, C. and Hussaini, M. (1988) Spectral Methods in Fluids Dynamics. Springer Series in Computational Physics, Springer-Verlag, Berlin. http://dx.doi.org/10.1007/978-3-642-84108-8

[6] Mitchell, A.R. and Griffiths, D.F. (1980) The Finite Difference Method in Partial Differential Equations. John Wiley & Sons, Chichester.

[7] Mickens, R.E. (2000) Applications of Nonstandard Finite Differences Schemes. World Scientific Publishing, River Edge.

[8] Dehghan, M. (2005) On the Numerical Solution of the One-Dimensional Convection-Diffusion Equation. Mathematical Problems in Engineering, 1, 61-74. http://dx.doi.org/10.1155/MPE.2005.61