Adaptive two‐step methods for semidiscretized wave equations

Abstract

SUMMARYAdaptive two-step direct integration methods are constructed for the integration of second-order semidisCrete evolution equations possessing oscillatory solutions. The methods are based on a class of adaptive multistep methods for a semilinear test model whose frequency is known. They are constructed following the notion of diagonally implicit RK-methods by using efficient rational approximations to cos v, v 2 0. Our interest is centered on the dispersion (or phase errors) of the dominant components in the numerical oscillations when these methods are applied to a linear homogeneous test model. Two-step methods which have high order of dispersion (up to 12), whereas the algebraic order is relatively low (2 or 4), are derived. Applications of these methods to linear as well as non-linear test models and to semidiscretized hyperbolic equations reveal a good behaviour with regard to error propagation when they are compared with other conventional methods.