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

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(iwt),exp(−iwt)} or equivalently the set {cos(wt),sin(wt)}, 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.