Linearly implicit Runge-Kutta methods with approximate matrix factorization can solve efficiently large systems of differential equations that have a stiff linear part, e.g. reaction-diffusion systems. However, the use of approximate factorization usually leads to loss of accuracy, which makes it attractive only for low order time integration schemes. This paper discusses the application of app...

in this paper, we intend to solve special kind of ordinary differential equations which is called heun equations, by converting to a corresponding stochastic differential equation(s.d.e.). so, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this s.d.e. is solved by numerically methods. mo...

Matlab has facilities for the numerical solution of ordinary differential equations (ODEs) of any order. In this document we first consider the solution of a first order ODE. Higher order ODEs can be solved using the same methods, with the higher order equations first having to be reformulated as a system of first order equations. Techniques for solving the first order and second order equation...

This paper is concerned with the approximation of the time domain Maxwell equations in a dispersive propagation medium by a Discontinuous Galerkin Time Domain (DGTD) method. The Debye model is used to describe the dispersive behaviour of the medium. We adapt the locally implicit time integration method from [1] and derive a convergence analysis to prove that the locally implicit DGTD method for...

High-order non-reflecting boundary conditions are introduced to create a finite computational space and for the solution of dispersive waves using a spectral element formulation with high-order time integration. Numerical examples are used to demonstrate the synergy of using high-order spatial, time, and boundary discretization. We show that by balancing all numerical errors involved, high-orde...

When pseudospectral approximations are used for space derivatives, one often encounters spurious eigenvalues. These can lead to severe time stepping difficulties for PDEs. This is especially the case for equations with high order derivatives in space, requiring multiple conditions at one or both boundaries. We note here that a very simple-to-implement fictitious point approach circumvents most ...

We investigate the behavior of adaptive time stepping numerical algorithms under the reverse mode of automatic differentiation (AD). By differentiating the time step controller and the error estimator of the original algorithm, reverse mode AD generates spurious adjoint derivatives of the time steps. The resulting discrete adjoint models become inconsistent with the adjoint ODE, and yield incor...

The dynamics of numerical approximation of cosymmetric ordinary differential equations with a continuous family of equilibria is investigated. Nonconservative and Hamiltonian model systems in two dimensions are considered and these systems are integrated with several firstorder Runge–Kutta methods. The preservation of symmetry and cosymmetry, the stability of equilibrium points, spurious soluti...

in this paper, we introduce a hybrid approach based on neural network and optimization teqnique to solve ordinary differential equation. in proposed model we use heyperbolic secont transformation function in hiden layer of neural network part and bfgs teqnique in optimization part. in comparison with existing similar neural networks proposed model provides solutions with high accuracy. numerica...

A new time integration strategy for the solution of convection-dominated partial differential equations in two space dimensions by the method of lines is presented. The strategy aims to ensure that the time integration error is less than the spatial discretisation error. This is achieved by making use of the individual contributions of the local spatial discretisation error and the local time i...