Euler-Chebyshev methods for integro-differential equations

We construct and analyse explicit methods for solving initial value problems for systems of differential equations with expensive righthand side functions whose Jacobian has its stiff eigenvalues along the negative axis. Such equations arise after spatial discretization of parabolic 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 righthand side evaluation per step and the construction of a stabilizing matrix. This matrix should be tuned to the class of problems to be integrated. In the case of parabolic integro-differential equations, the stabilizing matrix wil be based on Chebyshev polynomials and will be constructed by means of recursions satisfied by these polynomials. This construction is related to the construction of the intermediate stages in the so-called Runge-Kutta-Chebyshev methods for ordinary differential equations. In analogy with these methods, we shall call the stabilized Euler methods, Euler-Chebyshev methods. They are second-order accurate, and although they are explicit, their stepsize restriction is not prescribed by the stiff eigenvalues. For integro- differential equations in which the parabolic part consists of a one-dimensional diffusion term, we can describe an efficient implementation of the stabilizing matrix, which is based on factorization properties of Chebyshev polynomials.