In this paper, some simple families of explicit two-step finite difference methods for solving the wave equation in two and three spatial dimensions are examined. These schemes depend on several free parameters, and can be associated with so-called interpolated digital waveguide meshes. Special attention is paid to the stability properties of these schemes (in particular the bounds on the space...

Numerical simulation provides a effective tool for studying both the spatial and temporal nature of acoustic field on 3D or 4D timespace. The paper deals with the description of discrete exterior calculus scheme for the wave equation. This method can be directly implemented on manifold, which is the generation of finite difference time domain method from flat space to curved space.

In this paper, we consider the variable-order nonlinear fractional diffusion equation ∂u(x, t) ∂t = B(x, t)xR u(x, t) + f(u, x, t), where xR α(x,t) is a generalized Riesz fractional derivative of variable order α(x, t) (1 < α(x, t) ≤ 2) and the nonlinear reaction term f(u, x, t) satisfies the Lipschitz condition |f(u1, x, t) − f(u2, x, t)| ≤ L|u1 − u2|. A new explicit finite difference approxim...

In [Ler53] and [G̊ar56], Leray and Gårding have developed a multiplier technique for deriving a priori estimates for solutions to scalar hyperbolic equations in either the whole space or the torus. In particular, the arguments in [Ler53, G̊ar56] provide with at least one local multiplier and one local energy functional that is controlled along the evolution. The existence of such a local multipli...

High order integro-differential equations (IDE), especially nonlinear, are usually difficult to solve even for approximate solutions. In this paper, we give a high accurate compact finite difference method to efficiently solve integro-differential equations, including high order and nonlinear problems. By numerical experiments, we show that compact finite difference method of integro-differenti...

Based on multiplicative calculus, the finite difference schemes for the numerical solution of multiplicative differential equations and Volterra differential equations are presented. Sample problems were solved using these new approaches.

We consider the Bresse system with frictional dissipative terms acting in all the equations. We show the exponential decay of the solution by using a method developed by Z. Liu and S. Zheng and their collaborators in past years. The numerical computations were made by using the finite difference method to prove the theoretical results. In particular, the finite difference method in our case is ...

We use equations similar to the heat conduction equation to calulate heat transfer, radiation transfer and hydrostatical equilibrium in our stellar evolution programs. We tried various numerical schemes and found that the most convenient scheme for complicated calculations (nonlinear, multidimensional calculations) is a symmetrical semi-implicit (SSI) scheme. The (SSI) scheme is easy to code, v...

In the first part of this chapter we focus on the question of well-posedness of boundary-value problems for linear partial differential equations of elliptic type. The second part is devoted to the construction and the error analysis of finite difference schemes for these problems. It will be assumed throughout that the coefficients in the equation, the boundary data and the resulting solution ...

An explicit spectrally accurate order-adaptive Hermite-Taylor method for the Schrödinger equation is developed. Numerical experiments illustrating the properties of the method are presented. The method, which is able to use very coarse grids while still retaining high accuracy, compares favorably to an existing exponential integrator high order summation-by-parts finite difference method.