This paper presents a family of Runge{Kutta type integration schemes of arbitrarily high order for diierential equations evolving on manifolds. We prove that any classical Runge{Kutta method can be turned into an invariant method of the same order on a general homogeneous manifold, and present a family of algorithms that are relatively simple to implement.

In this paper we show that standard preconditioners for parabolic PDEs discretized by implicit Euler or Crank–Nicolson schemes can be reused for higher–order fully implicit Runge–Kutta time discretization schemes. We prove that the suggested block diagonal preconditioners are order–optimal for A–stable and irreducible Runge–Kutta schemes with invertible coefficient matrices. The theoretical inv...

In this paper, we analyze second-order Runge–Kutta approximations to a nonlinear optimal control problem with control constraints. If the optimal control has a derivative of bounded variation and a coercivity condition holds, we show that for a special class of Runge–Kutta schemes, the error in the discrete approximating control is O(h2) where h is the mesh spacing.

The non-linear wave equation is taken as a model problem for the investigation. Different multisymplectic reformulations of the equation are discussed. Multi-symplectic Runge–Kutta methods and multi-symplectic partitioned Runge–Kutta methods are explored based on these different reformulations. Some popular and efficient multi-symplectic schemes are collected and constructed. Stability analyses...

We prove that to every rational function R(z) satisfying R(−z)R(z) = 1, there exists a symplectic Runge-Kutta method with R(z) as stability function. Moreover, we give a surprising relation between the poles of R(z) and the weights of the quadrature formula associated with a symplectic Runge-Kutta method.

We are concerned with the solution of time-dependent nonlinear hyperbolic partial differential equations. We investigate the combination of residual distribution methods with a consistent mass matrix (discretisation in space) and a Runge-Kutta-type time stepping (discretisation in time). The introduced nonlinear blending procedure allows us to retain the explicit character of the time stepping ...

8. First-Order Equations: Numerical Methods 8.1. Numerical Approximations 2 8.2. Explicit and Implicit Euler Methods 3 8.3. Explicit One-Step Methods Based on Taylor Approximation 4 8.3.1. Explicit Euler Method Revisited 4 8.3.2. Local and Global Errors 4 8.3.3. Higher-Order Taylor-Based Methods (not covered) 5 8.4. Explicit One-Step Methods Based on Quadrature 6 8.4.1. Explicit Euler Method Re...

K e y w o r d s O r d i n a r y differential equations, Initial value problems, Runge-Kut ta methods, Stiffness detection, Symbolic computation, Computer algebra systems, Computer generation of numerical methods. 1. I N T R O D U C T I O N A framework for explicit Runge-Kutta methods is being implemented as part of an ongoing overhaul of MATHEMATICA~S differential equation solver NDSolve. One o...