We present a semi-implicit numerical method for micromagnetics simulations, in the presence of spin-currents. The dynamics of the magnetization are described by the Landau–Lifshitz–Gilbert equation, while the dynamics of the spin are described by a diffusion equation with discontinuous coefficients. The complexity of the method presented is comparable to that of solving the linear heat equation...

In contrast to stiff deterministic systems of ordinary differential equations, in general, the implicit Euler method for stiff stochastic differential equations is not effective. This paper introduces a new numerical method for stiff differential equations which consists of interlacing large implicit Euler time steps with a sequence of small explicit Euler time steps. We emphasize that uniform ...

we derive error estimates in the appropriate norms, for the streamlinediffusion (sd) finite element methods for steady state, energy dependent,fermi equation in three space dimensions. these estimates yield optimal convergencerates due to the maximal available regularity of the exact solution.high order sd method together with implicit integration are used. the formulationis strongly consistent...

Abstract. This paper focuses on two variants of the Milstein scheme, namely the split-step backward Milstein method and a newly proposed projected Milstein scheme, applied to stochastic differential equations which satisfy a global monotonicity condition. In particular, our assumptions include equations with super-linearly growing drift and diffusion coefficient functions and we show that both ...

Delay integro-differential equations are very important in biology, medicine and many other fields. If we take random noise into account, we can obtain many stochastic delay integro-differential equations (SDIDEs). As a special case of stochastic functional differential equations (SFDEs), the fundamental theory of existence and uniqueness of the solution of SDIDEs can be regarded similarly to t...

We discuss a leading-edge model used in the computation of the run-out length of dry-flowing avalanches. The model has the form of a singular initial value problem for a scalar ordinary differential equation describing the avalanche dynamics. Existence, uniqueness and smoothness properties of the analytical solution are shown. We also prove the existence of a unique root of the solution. Moreov...

A nonlinear degenerate convection-diffusion initial boundary value problem is studied in a bounded domain. A dynamical boundary condition (containing the time derivative of a solution) is prescribed on the one part of the boundary. This models a non-perfect contact on the boundary. Existence and uniqueness of a weak solution in corresponding function spaces is proved using the backward Euler me...

Numerical approximations to the solution of a linear singularly perturbed parabolic problem are generated using a backward Euler method in time and an upwinded finite difference operator in space on a piecewise-uniform Shishkin mesh for a convectiondiffusion problem. A proof is given to show first order convergence of these numerical approximations in appropriately weighted C-norm. Numerical re...

(a) Show that for the solution x(t, x0) of the IVP with initial value x0 holds: ‖x(t, x0)‖ = ‖x(t0, x0)‖ . What is the geometric interpretation? Hint: Differentiate 1/2 ‖x(t)‖ and try to formulate the derivative as a scalar product! (b) Formulate an IVP in C (i.e. for complex-valued functions) equivalent (?) to the one in R in (a). (c) Consider now the numerical approximation x∆ to the IVP, whe...

In this paper, two different numerical schemes, namely the Runge-Kutta fourth order method and the implicit Euler method with perturbation method of the second degree, are applied to solve the nonlinear thermal wave in one and two dimensions using the differential quadrature method. The aim of this paper is to make comparison between previous numerical schemes and detect which is more efficient...