High-order transverse schemes for the numerical solution of PDEs. (English) Zbl 0899.65055
Authors’ abstract: Many existing numerical schemes for the solution of initial-boundary value problems for partial differential equations (PDEs) can be derived by the method of lines. The PDEs are converted into a system of ordinary differential equations either with initial conditions (longitudinal scheme) or with boundary conditions (transverse scheme). In particular, this paper studies the performance of the transverse scheme in combination with boundary value methods. Moreover, we do not restrict the semi-discretization by the usual first- or second-order finite difference approximations to replace the derivative with respect to time, but we use high-order formulae.
Reviewer: M.Lénárd (Safat)
MSC:
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
| 35K15 | Initial value problems for second-order parabolic equations |
| 34B05 | Linear boundary value problems for ordinary differential equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
References:
| [1] | Amodio, P.; Brugnano, L., Parallel implementation of block boundary value methods for ODEs, J. Comput. Math. Appl., 78, 197-211 (1997) · Zbl 0868.65039 |
| [2] | Amodio, P.; Mazzia, F., A boundary value approach to the numerical solution of initial value problems by multisteps methods, J. Difference Equations Appl., 1, 353-367 (1995) · Zbl 0861.65062 |
| [3] | Amodio, P.; Paprzycki, M., Parallel solution of almost block diagonal systems on a hypercube, Linear Algebra Appl., 241/243, 85-103 (1996) · Zbl 0860.65066 |
| [4] | Ascher, U. M.; Mattheij, R. M.M.; Russel, R. D., Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (1988), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0666.65056 |
| [5] | Axelsson, A. O.H.; Verwer, J. G., Boundary value techniques for initial value problems in ordinary differential equations, Math. Comp., 45, 153-171 (1985) · Zbl 0586.65052 |
| [6] | Beam, R. M.; Warming, R. F., The asymptotic spectra of banded Toeplitz and quasi-Toeplitz matrices, SIAM J. Sci. Comput., 14, 971-1006 (1993) · Zbl 0788.65049 |
| [7] | Brugnano, L.; Trigiante, D., High order multistep methods for boundary value problems, Appl. Numer. Math., 18, 79-94 (1995) · Zbl 0837.65081 |
| [8] | Brugnano, L.; Trigiante, D., Convergence and stability of boundary value methods for ordinary differential equations, J. Comput. Appl. Math., 66, 97-109 (1996) · Zbl 0855.65087 |
| [9] | L. Brugnano, D. Trigiante, A new mesh selection strategy for ODEs, Appl. Numer. Math., to appear.; L. Brugnano, D. Trigiante, A new mesh selection strategy for ODEs, Appl. Numer. Math., to appear. · Zbl 0880.65066 |
| [10] | Carpenter, M. H.; Gottlieb, D.; Abarbanel, S., The stability of numerical boundary treatments for compact high-order finite-difference schemes, J. Comput. Phys., 108, 272-295 (1993) · Zbl 0791.76052 |
| [11] | Carpenter, M. H.; Gottlieb, D.; Abarbanel, S., Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes, J. Comput. Phys., 111, 220-236 (1994) · Zbl 0832.65098 |
| [12] | Cash, J. R., Stable Recursions (1979), Academic Press: Academic Press London · Zbl 0498.65035 |
| [13] | Diamantakis, M. T., The NUMOL solution of time-dependent PDEs using DESI Runge-Kutta formulae, Appl. Numer. Math., 17, 235-249 (1995) · Zbl 0832.65103 |
| [14] | Gustafsson, B.; Olsson, P., Fourth-order difference methods for hyperbolic IBVPs, J. Comput. Phys., 117, 300-317 (1995) · Zbl 0823.65083 |
| [15] | Hirsch, C., (Numerical Computation of Internal and External Flows, Vol. 1 (1988), Wiley: Wiley New York) · Zbl 0662.76001 |
| [16] | Iavernaro, F.; Mazzia, F., Convergence and stability of multistep methods solving nonlinear initial value problems, SIAM J. Sci. Computing, 18, 270-285 (1997) · Zbl 0870.65071 |
| [17] | Liskovets, O. A., The method of lines (review), Differential Equations, 1, 1308-1323 (1965) · Zbl 0229.65065 |
| [18] | Lopez, L.; Trigiante, D., Boundary value methods and BV-stability in the solution of initial value problems, Appl. Numer. Math., 11, 225-239 (1993) · Zbl 0789.65054 |
| [19] | Marzulli, P.; Trigiante, D., On some numerical methods for solving ODE, J. Difference Equations Appl., 1, 45-55 (1995) · Zbl 0834.65081 |
| [20] | Mazzia, F., Boundary value methods for initial value problems: parallel implementation, Ann. Numer. Math., 1, 439-450 (1994) · Zbl 0828.65077 |
| [21] | Miller, J. J.H., On the stability of differential equations, SIAM J. Control, 19, 639-648 (1972) · Zbl 0256.34059 |
| [22] | Mitchell, A. R.; Griffiths, D. F., The finite difference method in partial differential equations (1980), Wiley: Wiley Chichester · Zbl 0417.65048 |
| [23] | Olver, F. W.J., Numerical solutions of second order linear difference equations, J. Res. Nat. Bur. Standards Math. Math. Phys., 71B, 2, 3, 111-129 (1967) · Zbl 0171.36601 |
| [24] | Reiter, J., A biplicit spectral-collocation-type ansatz for the numerical integration of partial differential equations with the transversal method of lines, Numer. Methods Partial Differential Equations, 11, 625-635 (1995) · Zbl 0836.65102 |
| [25] | Sanz-Serna, J. M.; Verwer, J. G.; Hundsdorfer, W. H., Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations, Num. Mat., 50, 405-418 (1986) · Zbl 0589.65069 |
| [26] | Schiesser, W. E., The Numerical Method of Lines (1991), Academic Press: Academic Press New York · Zbl 0763.65076 |
| [27] | Shampine, L. F., ODE solvers and the method of lines, Numer. Methods Partial Differential Equations, 10, 739-755 (1994) · Zbl 0826.65082 |
| [28] | Verwer, J. G.; Sanz-Serna, J. M., Convergence of method of lines approximations to partial differential equations, Computing, 33, 297-313 (1984) · Zbl 0546.65064 |
| [29] | Wright, S., Stable parallel algorithms for two-point boundary value problems, SIAM J. Sci. Statist. Comput., 13, 742-764 (1992) · Zbl 0757.65095 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
