EULER-CHEBYSHEV METHODS FOR INTEGRODIFFERENTIAL EQUATIONS

We construct and analyse explicit methods for solving initial value pr oblems for systems of differential equations with expensive right-hand side functions whose Jacobian has its stiff eigenvalues along the neg ative axis. Such equations arise after spatial discretization of parab olic integro-differential equations of Volterra or Fredholm type with nonstiff integral parts, The methods to be developed in this paper may be interpreted as stabilized forward Euler methods. They require only one right-hand side evaluation per step and the construction of a sta bilizing matrix, This matrix should be tuned to the class of problems to be integrated, In the case of parabolic integro-differential equati ons, the stabilizing matrix will be based on Chebyshev polynomials and will be constructed by means of recursions satisfied by these polynom ials. This construction is related to the construction of the intermed iate stages in the so-called Runge-Kutta-Chebyshev methods for ordinar y differential equations. In analogy with these methods, we shall call the stabilized Euler methods, Euler-Chebyshev methods. They are secon d-order accurate, and although they are explicit, their stepsize restr iction is not prescribed by the stiff eigenvalues, For integro-differe ntial equations in which the parabolic part consists of a one-dimensio nal diffusion term, we describe an efficient implementation of the sta bilizing matrix, which is based on factorization properties of Chebysh ev polynomials. (C) 1997 Published by Elsevier Science B.V.