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A MATLAB implementation of upwind finite differences and adaptive grids in the method of lines. (English) Zbl 1071.65544
Summary: We report on the development of a MATLAB library for the solution of partial differential equation systems following the method of lines. In particular, we focus attention on upwind finite difference schemes and grid adaptivity, i.e., grid movement or grid refinement. Several algorithms are presented and their performance is demonstrated with illustrative examples including a fixed-bed reactor with periodic flow reversal, a model of flame propagation, and the Korteweg-de Vries equation.

MSC:
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
65Y15 Packaged methods for numerical algorithms
35Q53 KdV equations (Korteweg-de Vries equations)
80A25 Combustion
80M20 Finite difference methods applied to problems in thermodynamics and heat transfer
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[1] Berzins, M., Developments in NAG library software for parabolic equations, () · Zbl 0888.65109
[2] Blom, J.G.; Zegeling, P.A., Algorithm 731: a moving-grid interface for systems of one-dimensional time-dependent partial differential equations, ACM trans. math. software, 20, 194-214, (1994) · Zbl 0889.65099
[3] H.A. Dwyer, B.R. Sanders, Numerical modeling of unsteady flame propagation, Sandia National Laboratory Livermore Report SAND77-8275, 1978. · Zbl 0388.76090
[4] Eigenberger, G.; Nieken, C., Catalytic combustion with periodic flow reversal, Chem. eng. sci., 43, 2109-2115, (1988)
[5] Fornberg, B., Calculation of weights in finite difference formulas, SIAM rev., 40, 685-691, (1998) · Zbl 0914.65010
[6] Kautsky, J.; Nichols, N.K., Equidistributing meshes with constraints, SIAM J. sci. stat. comput., 1, 499-511, (1980) · Zbl 0455.65068
[7] Korteweg, D.J.; de Vries, G., On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Philos. mag., 39, 422-443, (1895) · JFM 26.0881.02
[8] Pennington, S.V.; Berzins, M., New NAG library software for first-order systems of time-dependent pdes, ACM trans. math. software, 20, 63, (1994) · Zbl 0888.65109
[9] Saucez, P.; Schiesser, W.E.; Vande Wouwer, A., Upwinding in the method of lines, Math. comput. simulat., 56, 171-185, (2001) · Zbl 0977.68022
[10] Saucez, P.; Vande Wouwer, A.; Schiesser, W.E., Some observations on a static spatial remeshing method based on equidistribution principles, J. comput. phys., 128, 274-288, (1996) · Zbl 0859.65098
[11] P. Saucez, A. Vande Wouwer, P. Zegeling, Adaptive mesh computations for the extended fifth-order Korteweg – de Vries equation, J. Comput. Appl. Math., this issue doi:10.1016/j.cam.2004.12.028. · Zbl 1075.35072
[12] Schiesser, W.E., The numerical method of lines: integration of partial differential equations, (1991), Academic Press San Diego · Zbl 0763.65076
[13] Shampine, L.F.; Gladwell, I.; Thompson, S., Solving ODEs with MATLAB, (2003), Cambridge University Press Cambridge · Zbl 1079.65144
[14] Trefethen, L.N., Spectral methods in MATLAB, (2000), SIAM Philadelphia · Zbl 0953.68643
[15] A. Vande Wouwer, Ph. Saucez, W.E. Schiesser (Eds.), Adaptive Method of Lines, Chapman & Hall/CRC, Boca Raton, 2001. · Zbl 0986.65083
[16] Vande Wouwer, A.; Saucez, Ph.; Schiesser, W.E., Simulation of distributed parameter systems using a MATLAB-based method of lines toolbox—chemical engineering applications, Ind. eng. chem. res., 43, 3469-3477, (2004)
[17] Verwer, J.G.; Blom, J.G.; Furzeland, R.M.; Zegeling, P.A., A moving-grid method for one-dimensional PDEs based on the method of lines, (), 160-175 · Zbl 0697.65086
[18] Weideman, J.A.C.; Reddy, S.C., A MATLAB differentiation matrix suite, ACM trans. math. software, 26, 465-519, (2000)
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