This paper simulates transient compressible adiabatic gas flows in a long pipeline following a catastrophic failure using an implicit high order finite difference scheme as a discretisation technique for convective terms. A rupture is assumed to be occurred accidentally at a distance from the feeding point of a pipeline where the pipeline being divided into two distinct sections, high and low p...

In this paper we present a random walk model for approximating a Lévy-Feller advection-dispersion process, governed by the Lévy-Feller advection-dispersion differential equation (LFADE). We show that the random walk model converges to LFADE by use of a properly scaled transition to vanishing space and time steps. We propose an explicit finite difference approximation (EFDA) for LFADE, resulting...

We put forward and analyze an explicit finite difference scheme for the Camassa-Holm shallow water equation that can handle general H1 initial data and thus peakon-antipeakon interactions. Assuming a specified condition restricting the time step in terms of the spatial discretization parameter, we prove that the difference scheme converges strongly in H1 towards a dissipative weak solution of C...

A method for deriving reduced dynamic models of one-dimensional distributed systems is presented. It inherits the concepts of the aggregated modeling method of Lévine and Rouchon originally derived for simple staged distillation models and can be applied to both spatially discrete and continuous systems. The method is based on partitioning the system into intervals of steady-state systems, whic...

In this paper we propose a finite difference scheme for temporal discretization of the time-fractional thermistor problem, which is obtained from the so-called thermistor problem by replacing the first-order time derivative with a fractional derivative of order α (0 ≤ α ≤ 1). An existence result is established for the semidiscrete problem. Stability and error analysis are then provided, showing...

We present a finite difference method for discretizing a Heaviside function H(u(~x)), where u is a level set function u : Rn 7→ R that is positive on a bounded region Ω ⊂ R. There are two variants of our algorithm, both of which are adapted from finite difference methods that we proposed for discretizing delta functions in [13–15]. We consider our approximate Heaviside functions as they are use...

A class of finite difference methods for solving fractional diffusion equations is considered. These methods are an extension of the weighted average methods for ordinary (non-fractional) diffusion equations. Their accuracy is of order (Dx) and Dt, except for the fractional version of the Crank–Nicholson method, where the accuracy with respect to the timestep is of order (Dt) if a second-order ...

where A : D(A) ⊆H →H , α : D(α) ⊆H →H , and β : D(β) ⊆H →H are maximal monotone operators in the real Hilbert space H (satisfying some specific properties), a, b are given elements in the domain D(A) of A, f ∈ L2(0,T ;H), and p,r : [0,T] → R are continuous functions, p(t) ≥ k > 0 for all t ∈ [0,T]. Particular cases of this problem were considered before in [9, 10, 12, 15, 16]. If p ≡ 1, r ≡ 0, ...

We consider the numerical approximation of singularly perturbed linear second order reaction-diffusion boundary value problems with a small shift(δ) in the undifferentiated term and the shift depends on the small parameter(ǫ). The presence of small parameter induces twin boundary layers. The problem is discretized using standard finite difference scheme on an uniform mesh and the retarded argum...

In this paper, a weak formulation of the discontinuous variable coefficient Poisson equation with interfacial jumps is studied. The existence, uniqueness and regularity of solutions of this problem are obtained. It is shown that the application of the Ghost Fluid Method by Fedkiw, Kang, and Liu to this problem in [9] can be obtained in a natural way through discretization of the weak formulatio...