Symmetric trigonometrically-fitted two-step hybrid methods for oscillatory problems

Abstract

In this paper, we propose a general class of trigonometrically-fitted two-step hybrid (TFTSH) methods for solving numerically oscillatory second-order initial value problems. The TFTSH methods integrate exactly the differential system whose solutions can be expressed as the linear combinations of functions from the set { exp ( i w t ) , exp ( − i w t ) } or equivalently the set { cos ( w t ) , sin ( w t ) } , where w represents an approximation of the main frequency of the problem. By introducing the generalized B2-series, the necessary and sufficient conditions for TFTSH methods of up to arbitrarily high order p are derived. We also investigate the symmetry of TFTSH methods and analyze the symmetric conditions of TFTSH methods. Based on the order conditions and symmetric conditions, a diagonally-implicit two-stage symmetric TFTSH method with order four is constructed. Some numerical experiments are provided to confirm the theoretical expectations. • A general class of trigonometrically-fitted two-step hybrid (TFTSH) methods is proposed. • General order conditions are derived by introducing the generalized B2-series. • The symmetry of the new methods is investigated and the symmetric conditions are given. • A diagonally implicit two-stage symmetric TFTSH method is given. • Numerical results show the efficiency of our new method.