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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Generalized finite-difference schemes


Authors: Blair Swartz and Burton Wendroff
Journal: Math. Comp. 23 (1969), 37-49
MSC: Primary 65.65
MathSciNet review: 0239768
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Abstract: Finite-difference schemes for initial boundary-value problems for partial differential equations lead to systems of equations which must be solved at each time step. Other methods also lead to systems of equations. We call a method a generalized finite-difference scheme if the matrix of coefficients of the system is sparse. Galerkin's method, using a local basis, provides unconditionally stable, implicit generalized finite-difference schemes for a large class of linear and nonlinear problems. The equations can be generated by computer program. The schemes will, in general, be not more efficient than standard finite-difference schemes when such standard stable schemes exist. We exhibit a generalized finite-difference scheme for Burgers' equation and solve it with a step function for initial data.


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DOI: http://dx.doi.org/10.1090/S0025-5718-1969-0239768-7
PII: S 0025-5718(1969)0239768-7
Article copyright: © Copyright 1969 American Mathematical Society