Higher order compact implicit schemes for the wave equation
HTML articles powered by AMS MathViewer
- by Melvyn Ciment and Stephen H. Leventhal PDF
- Math. Comp. 29 (1975), 985-994 Request permission
Abstract:
Higher order finite-difference techniques have associated large star systems which engender complications near the boundary. In the numerical solution of hyperbolic equations, such boundary conditions require careful treatment since errors or instabilities generated there will in general pollute the entire calculation. To circumvent this difficulty, we use a class of implicit schemes suggested by H.-O. Kreiss, which achieves the highest order of accuracy possible on the smallest (most compact) mesh system. Here we develop a scheme which approximates the wave equation, \[ {U_{tt}} = a(x,y,t){U_{xx}} + b(x,y,t){U_{yy}},\] with fourth order accuracy in space and time. After an appropriate factorization, the resulting set of equations are tridiagonal and hence easily solved. The tridiagonal nature also indicates that the boundary conditions do not create special difficulties. Numerical experiments demonstrate the expected order of convergence and fulfill our expectations on the treatment of boundary conditions. An experimental computation also demonstrates that our results hold on L-shaped domains.References
- G. Fairweather and A. R. Mitchell, A high accuracy alternating direction method for the wave equation, J. Inst. Math. Appl. 1 (1965), 309–316. MR 210331
- Eugene Isaacson and Herbert Bishop Keller, Analysis of numerical methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0201039
- Milton Lees, Alternating direction methods for hyperbolic differential equations, J. Soc. Indust. Appl. Math. 10 (1962), 610–616. MR 172475 S. McKEE, "High accuracy A.D.I. methods for hyperbolic equations with variable coefficients," J. Inst. Math. Appl., v. 11, 1973, pp. 105-109. S. A. ORSZAG & M. ISRAELI, "Numerical simulation of viscous incompressible flows," Annual Review of Fluid Mechanics. Vol. VI, M. Van Dyke (Editor), Annual Reviews, 1974, pp. 281-318.
- David M. Young, Iterative solution of large linear systems, Academic Press, New York-London, 1971. MR 0305568
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Math. Comp. 29 (1975), 985-994
- MSC: Primary 65M05; Secondary 35L05
- DOI: https://doi.org/10.1090/S0025-5718-1975-0416049-2
- MathSciNet review: 0416049