Rooted tree analysis of Runge-Kutta methods with exact treatment of linear terms

We investigate a class of time discretization schemes called “ETD Runge-Kutta methods,” where the linear terms of an ordinary differential equation are treated exactly, while the other terms are numerically integrated by a one-step method. These schemes, proposed by previous authors, can be regarded as modified Runge-Kutta methods whose coefficients are matrices instead of scalars. From this viewpoint, we reexamine the notion of consistency, convergence and order to provide a mathematical foundation for new methods. Applying the rooted tree analysis, expansion theorems of both the strict and numerical solutions are proved, and two types of order conditions are defined. Several classes of formulas with up to four stages that satisfy the conditions are constructed, and it is shown that the power series of matrices, employed as their coefficients, can be determined using the order conditions.