The conditioning of implicit Runge–Kutta (RK) integration for linear finite element approximation diffusion equations on general anisotropic meshes is investigated. Bounds are established the condition number resulting system with and without diagonal preconditioning Euler (the simplest RK method) methods. Two solution strategies considered from integration: simultaneous where solved as a whole...

Implicit/explicit (IMEX) Runge-Kutta (RK) schemes are effective for time-marching ODE systems with both stiff and nonstiff terms on the RHS; such schemes implement an (often A-stable or better) implicit RK scheme for the stiff part of the ODE, which is often linear, and, simultaneously, a (more convenient) explicit RK scheme for the nonstiff part of the ODE, which is often nonlinear. Low-storag...

In this paper, we consider the problem of computing numerical solutions for stochastic differential equations (SDEs) of Itô form. A fully explicit method, the split-step forward Milstein (SSFM) method, is constructed for solving SDEs. It is proved that the SSFM method is convergent with strong order γ = 1 in the mean-square sense. The analysis of stability shows that the mean-square stability p...

A two-dimensional nonhydrostatic (NH) atmospheric model based on the compressible Euler system has been developed in the (x, z) Cartesian domain. The spatial discretization is based on a nodal discontinuous Galerkin (DG) method with exact integration. The orography is handled by the terrain-following heightbased coordinate system. The time integration uses the horizontally explicit and vertical...

We consider the numerical solution of systems of semi-explicit index 2 differential-algebraic equations (DAEs) by methods based on RungeKutta (RK) coefficients. For nonstiffly accurate RK coefficients, such as Gauss and Radau IA coefficients, the standard application of implicit RK methods is generally not superconvergent. To reestablish superconvergence projected RK methods and partitioned RK ...

This paper describes how the specific access structure of the Brusselator equation, a typical example for ordinary differential equations (ODEs) derived by the method of lines, can be exploited to obtain scalable distributed-memory implementations of explicit Runge-Kutta (RK) solvers. These implementations need less communication and therefore achieve better speed-ups than general explicit RK i...

We construct optimal k-step, 5to 10-stage, explicit, strong-stability-preserving Hermite-Birkhoff (SSP HB) methods of order 12 with nonnegative coefficients by combining linear k-step methods of order 9 with 5to 10-stage Runge-Kutta (RK) methods of order 4. Since these methods maintain the monotonicity property, they are well suited for solving hyperbolic PDEs by the method of lines after a spa...