In this paper, we unify advection and diffusion into a single hyperbolic system by extending the firstorder system approach introduced for the diffusion equation in [J. Comput. Phys., 227 (2007) 315-352] to the advection-diffusion equation. Specifically, we construct a unified hyperbolic advection-diffusion system by expressing the diffusion term as a first-order hyperbolic system and simply ad...

This note recounts detailed stability and accuracy analysis of a scalar advection-diffusion equation.

We design a perfectly matched layer for the advection-diffusion equation. We show that the reflection coefficient is exponentially small with respect to the damping parameter and the width of the PML and this independently of the advection and of the viscosity. Numerical tests assess the efficiency of the approach.

Subdiffusive fractional equations are not structurally stable with respect to spatial perturbations to the anomalous exponent (Phys. Rev. E 85, 031132 (2012)). The question arises of applicability of these fractional equations to model real world phenomena. To rectify this problem we propose the inclusion of the random death process into the random walk scheme from which we arrive at the modifi...

We develop an optimization-based approach for additive decomposition and reconnection of algebraic problems arising from discretization of partial differential equations (PDEs). Application to a scalar advection-diffusion PDE illustrates the new approach. In particular, we obtain a robust iterative solver for advection-dominated problems using standard multi-level solvers for the Poisson equation.

Most of this chapter is devoted to a discussion of the one-dimensional advection equation, (4.1) Here A is the advected quantity, and c is the advecting current. This is a linear, first-order, partial differential equation with a constant coefficient, namely c. Both space and time differencing are discussed in this chapter, but more emphasis is placed on space differencing. We have already pres...

A transport equation that uses fractional-order dispersion derivatives has fundamental solutions that are Lévy’s a-stable densities. These densities represent plumes that spread proportional to time, have heavy tails, and incorporate any degree of skewness. The equation is parsimonious since the dispersion parameter is not a function of time or distance. The scaling behavior of plumes that unde...

in this paper, a high-order and conditionally stable stochastic difference scheme is proposed for the numerical solution of $rm ithat{o}$ stochastic advection diffusion equation with one dimensional white noise process. we applied a finite difference approximation of fourth-order for discretizing space spatial derivative of this equation. the main properties of deterministic difference schemes,...

A mathematical method called subordination broadens the applicability of the classical advection--dispersion equation for contaminant transport. In this method, the time variable is randomized to represent the operational time experienced by different particles. In a highly heterogeneous aquifer, the operational time captures the fractal properties of the medium. This leads to a simple, parsimo...

Recent work on the global overturning circulation and its energetics assumes that processes caused by nonlinearities of the equation of state of seawater are negligible. Nonlinear processes such as cabbeling and thermobaricity cause diapycnal motion as a consequence of isopycnal mixing. The nonlinear equation of state also causes the helical nature of neutral trajectories; as a consequence of t...