Exponentially Fitted - Diagonally Implicit Runge-Kutta Method for Direct Solution of Fifth-Order Ordinary Differential Equations

In this paper, an Exponentially fitted - Diagonally implicit Runge-Kutta method is constructed, which can solve fifth-order ordinary differential equations (ODEs) directly. The order conditions are calculated using the expansion of the B-string theory and the colored tree theory to determine the ranking criteria of the Diagonally Implicit Runge-Kutta Method (DIRKF) approach. As a result, a five-degree, three-stage exponentially fitted - diagonally implicit Runge-Kutta method (EFDIRKFO5) is formulated. Comparing this method with existing implicit Runge-Kutta methods, numerical experiments show that the former is more accurate and requires fewer function evaluations.