I present a numerical solution of linear and nonlinear stiff problems using the RK-Butcher algorithm. The obtained discrete solutions using the RK-Butcher algorithm are found to be very accurate and are compared with the exact solutions of the linear and nonlinear stiff problems and also with the Runge-Kuttamethod based on arithmeticmean (RKAM). A topic of stability for the RK-Butcher algorithm...

A new block extended backward differentiation formula suitable for the integration of stiff initial value problems is derived. The procedure used involves the use of an extra future point which helps in improving the performance of an existing block backward differentiation formula. The method approximates the solution at 3 points simultaneously at each step. Accuracy and stability properties o...

In this work we present finite element approximations of relaxed systems for nonlinear diffusion problems, which can also tackle the cases of degenerate and strongly degenerate diffusion equations. Relaxation schemes take advantage of the replacement of the original partial differential equation (PDE) with a semilinear hyperbolic system of equations, with a stiff source term, tuned by a relaxat...

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration ordinary differential equations (ODEs) is presented. Its derivation based on integral form equation. The approach enables enhancing accuracy established Runge–Kutta while retaining same number stages. We demonstrate that, with proposed approach, Gauss–Legendre and Lobatto IIIA can be derived that th...

Despite the popularity of high-order explicit Runge–Kutta (ERK) methods for integrating semi-discrete systems of equations, ERK methods suffer from severe stability-based time step restrictions for very stiff problems. We implement a discontinuous Galerkin finite element method (DGFEM) along with recently introduced high-order implicit–explicit Runge–Kutta (IMEX-RK) schemes to overcome geometry...

This paper reviews various aspects of stiffness in the numerical solution of initial-value problems for systems of ordinary differential equations. In the literature on numerical methods for solving initial value problems the term "stiff" has been used by various authors with quite different meanings, which often causes confusion. This paper attempts to clear up this confusion by reviewing some...

In this article we propose a new class of finite volume schemes of arbitrary accuracy in space and time for systems of hyperbolic balance laws with stiff source terms. The new class of schemes is based on a three stage procedure. First, in order to achieve high order accuracy in space, a nonlinear weighted essentially non-oscillatory reconstruction procedure is applied to the cell averages at t...

Among the most popular methods for the solution of the Initial Value Problem are the Runge–Kutta pairs of orders 5 and 4. These methods can be derived solving a system of nonlinear equations for its coefficients. For achieving this, we usually admit various simplifying assumptions. The most common of them are the so called row simplifying assumptions. Here we negligible them and present an algo...

We consider implicit-explicit (IMEX) Runge Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strong-stabilitypreserving (SSP) scheme, and the implicit part is treated by an L-stable diagonally implicit Runge Kutta (DIRK). The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit. High accuracy in space...

In this paper, we develop bound-preserving modified exponential Runge-Kutta (RK) discontinuous Galerkin (DG) schemes to solve scalar conservation laws with stiff source terms by extending the idea in Zhang and Shu [39]. Exponential strong stability preserving (SSP) high order time discretizations are constructed and then modified to overcome the stiffness and preserve the bound of the numerical...