Abstract
Tidal forcing of the shallow water equations is typical of a class of problems where an approximate equilibrium solution is sought by long time integration of a differential equation system. A combination of the angled-derivative scheme with a staggered leap-frog scheme is sometimes used to discretise this problem. It is shown here why great care then needs to be taken with the boundary conditions to ensure that spurious solution modes do not lead to numerical instabilities. Various techniques are employed to analyse two simple model problems and display instabilities met in practical computations; these are then used to deduce a set of stable boundary conditions.
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References
R. Beam and R. F. Warming, An implicit finite-difference algorithm for hyperbolic systems in conservation law form, J. Comput. Phys. 22 (1976) 87–110.
N. Burgess, Stable boundary conditions for the shallow water equations, Ph.D. thesis, Oxford University Computing Laboratory (1987).
N. A. Edwards, C. P. Please and R. W. Preston, Some observations on boundary conditions for the shallow water equations in two space dimensions, IMA J. Numer. Anal. 30 (1983) 161–172.
N. A. Edwards and R. W. Preston, Grid scale oscillation in FLOW — the C.E.R.L. shallow water solver, C.E.R.L. Report TPRD/L/2779/N84 (1984).
M. G. Foreman, An accuracy analysis of boundary conditions for the forced shallow water equations, J. Comput. Phys. 64 (1986) 334.
S. K. Godunov and V. S. Ryabenkii,The Theory of Difference Schemes — An Introduction (North-Holland, Amsterdam, 1964).
B. Gustafsson, H.-O. Kreiss and A. Sundström, Stability theory of difference approximations for mixed initial boundary value problems II, Math. Comp. 26(119) (1972) 649–686.
J. G. Heywood and R. Rannacher, Finite element approximations of the Navier-Stokes problem. Part II: Stability of solutions and error estimates uniform in time, SIAM J. Numer. Anal. 23 (1966) 750–777.
R. L. Higdon, Initial-boundary value problems for linear hyperbolic systems, SIAM Rev. 28(2) (1986) 177–217.
K. W. Morton, Initial-value problems by finite difference and other methods, in:The State of the Art in Numerical Analysis, Proceedings of the IMA Conference, York, 1976, ed. D. A. H. Jacobs (Academic Press, 1977) pp. 699–756.
R. D. Richtmyer and K. W. Morton,Difference Methods for Initial-Value Problems (Wiley-Interscience, 2nd ed., 1967); reprint edition (Krieger Publishing Company, Malabar, 1994).
K. V. Roberts and N. O. Weiss, Convective difference schemes, Math. Comp. 20(94) (1966) 272–299.
G. Strang, Wiener-Hopf difference equations, J. Math. Mech. 13(1) (1964) 85.
G. Strang, Implicit difference methods for initial-boundary value problems, J. Math. Anal. Appl. 16 (1966) 188.
L. N. Trefethen, Group velocity interpretation of the stability theory of Gustafsson, Kreiss, and Sundström, J. Comput. Phys. 49 (1983) 199–217.
S. Wardrop, The computation of equilibrium solutions of forced hyperbolic partial differential equations, Ph.D. thesis, Oxford University Computing Laboratory (1990).
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Dedicated to Professor J. Crank on the occasion of his 80th birthday
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Morton, K.W., Burgess, N.A. The stability of boundary conditions for an angled-derivative difference scheme. Adv Comput Math 6, 263–279 (1996). https://doi.org/10.1007/BF02127707
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DOI: https://doi.org/10.1007/BF02127707
Keywords
- numerical stability
- Godunov-Ryabenkii conditions
- initial boundary value problem
- angled-derivative difference scheme
- tidally-forced shallow water equations