Publications of the Research Institute for Mathematical Sciences
A method for exponential propagation of large systems of stiff nonlinear differential equations
Journal of Scientific Computing
Krylov methods for the incompressible Navier-Stokes equations
Journal of Computational Physics
A new class of time discretization schemes for the solution of nonlinear PDEs
Journal of Computational Physics
Pseudo-Runge-Kutta Methods Involving Two Points
Journal of the ACM (JACM)
Spectral methods in MatLab
A quasi-steady state solver for the stiff ordinary differential equations of reaction kinetics
Journal of Computational Physics
An exponential method of numerical integration of ordinary differential equations
Communications of the ACM
Exponential time differencing for stiff systems
Journal of Computational Physics
Additive Runge-Kutta schemes for convection-diffusion-reaction equations
Applied Numerical Mathematics
Adaptive solution of partial differential equations in multiwavelet bases
Journal of Computational Physics
Fourth-Order Time-Stepping for Stiff PDEs
SIAM Journal on Scientific Computing
Generalized integrating factor methods for stiff PDEs
Journal of Computational Physics
Journal of Computational and Applied Mathematics
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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.