Degenerate Parabolic Convection Research Articles

A parameter uniform numerical method on a Bakhvalov type mesh for singularly perturbed degenerate parabolic convection–diffusion problems

AbstractWe are focused on the numerical treatment of a singularly perturbed degenerate parabolic convection–diffusion problem that exhibits a parabolic boundary layer. The discretization and analysis of the problem are done in two steps. In the first step, we discretize in time and prove its uniform convergence using an auxiliary problem. In the second step, we discretize in space using an upwind scheme on a Bakhvalov-type mesh and prove its uniform convergence using the truncation error and barrier function approach, wherein several bounds derived for the mesh step sizes are used. Numerical results for a couple of examples are presented to support the theoretical bounds derived in the paper.

A second order fitted operator finite difference scheme for a singularly perturbed degenerate parabolic problem

A second order finite difference scheme is constructed to solve a singularly perturbed degenerate parabolic convection diffusion problem via Rothe's method. The solution of the problem exhibits a boundary layer on the left side of the spatial domain. By means of the Crank Nicolson finite difference scheme, the time derivative is discretised to obtain a set of semi-discrete boundary value problems. Using a fitted operator finite difference scheme based on the midpoint downwind scheme, the system of boundary value problems are discretized and analysed for convergence. Second order accuracy is established for each discretisation process. Numerical simulations are carried out to validate the theoretical error estimate.

A parameter-uniform grid equidistribution method for singularly perturbed degenerate parabolic convection–diffusion problems

We consider a class of singularly perturbed degenerate parabolic convection–diffusion problems on a rectangular domain. A numerical method is constructed using the implicit Euler scheme on a uniform mesh in the time direction and the upwind finite difference scheme on a layer adaptive non-uniform mesh in the spatial direction. The layer adaptive non-uniform mesh in the spatial direction is generated through the equidistribution of a suitably chosen monitor function. We perform error analysis through the truncation error and barrier function approach and prove that the method is uniformly convergent with first order in both time and space. Numerical results are given in support of theoretical findings.

Computational uncertainty quantification for some strongly degenerate parabolic convection–diffusion equations

Strongly degenerate parabolic convection–diffusion equations arise as governing equations in a number of applications such as traffic flow with driver reaction and anticipation distance and sedimentation of solid–liquid suspensions in mineral processing and wastewater treatment. In these applications several parameters that define the convective flux function and the degenerating diffusion coefficient are subject to stochastic variability. A method to evaluate the variability of the solution of the governing partial differential equation in response to that of the parameters is presented. To this end, a generalized polynomial chaos (gPC) expansion of the solution is approximated by its projection onto a finite-dimensional space of piecewise polynomial functions defined on a suitable discretization of the stochastic domain, according to the basic principle of the hybrid stochastic Galerkin (HSG) approach. This approach is combined with a finite volume (FV) method, resulting in a so-called FV–HSG method, to compute the sought deterministic coefficient functions of the truncated polynomial-chaos-based expansion of the solution. Since the stochastic parameter space is now spanned by piecewise polynomial functions, one may employ the numerical result to compute the reconstruction of the numerical solution for arbitrary values of the random variables. The expectation, the variance or other stochastic quantities of the solution (as functions of time and position) can also be computed from these coefficient functions. The method is illustrated by a number of numerical examples.

An ϵ-uniform hybrid numerical scheme for a singularly perturbed degenerate parabolic convection–diffusion problem

ABSTRACTIn this paper, we study the numerical solution of singularly perturbed degenerate parabolic convection–diffusion problem on a rectangular domain. The solution of the problem exhibits a parabolic boundary layer in the neighbourhood of x=0. First, we use the backward-Euler finite difference scheme to discretize the time derivative of the continuous problem on uniform mesh in the temporal direction. Then, to discretize the spatial derivatives of the resulting time semidiscrete problem, we apply the hybrid finite difference scheme, which is a combination of central difference scheme and midpoint upwind scheme on piecewise uniform Shishkin mesh. We derive the error estimates, which show that the proposed hybrid scheme is ϵ-uniform convergent of almost second-order (up to a logarithmic factor) in space and first-order in time. Some numerical results have been carried out to validate the theoretical results.

Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition

We study a scalar conservation law whose flux has a single spatial discontinuity. There are many notions of (entropy) solution, the relevant concept being determined by the application. We focus on the so-called vanishing viscosity solution. We utilize a Kružkov-type entropy inequality which generalizes the one in [K. H. Karlsen, N. H. Risebro and J. D. Towers, [Formula: see text]-stability for entropy solutions of nonlinear degenerate parabolic convection–diffusion equations with discontinuous coefficients, Skr. K. Nor. Vidensk. Selsk. 3 (2003) 1–49], singles out the vanishing viscosity solution whether or not the crossing condition is satisfied, and has a discrete version satisfied by the Godunov variant of the finite difference scheme of [S. Diehl, On scalar conservation laws with point source and discontinuous flux function, SIAM J. Math. Anal. 26(6) (1995) 1425–1451]. We show that the solutions produced by that scheme converge to the unique vanishing viscosity solution. The scheme does not require a Riemann solver for the discontinuous flux problem. This makes its implementation simple even when the flux is multimodal, and there are multiple flux crossings.

A multilevel Monte Carlo finite difference method for random scalar degenerate convection–diffusion equations

This paper proposes a finite difference multilevel Monte Carlo algorithm for degenerate parabolic convection–diffusion equations where the convective and diffusive fluxes are allowed to be random. We establish a notion of stochastic entropy solutions to these equations. Our chief goal is to efficiently compute approximations to statistical moments of these stochastic entropy solutions. To this end, we design a multilevel Monte Carlo method based on a finite volume scheme for each sample. We present a novel convergence rate analysis of the combined multilevel Monte Carlo finite volume method, allowing in particular for low [Formula: see text]-integrability of the random solution with [Formula: see text], and low deterministic convergence rates (here, the theoretical rate is [Formula: see text]). We analyze the design and error versus work of the multilevel estimators. We obtain that the maximal rate (based on optimizing possibly the pessimistic upper bounds on the discretization error) is obtained for [Formula: see text], for finite volume convergence rate of [Formula: see text]. We conclude with numerical experiments.

Second-Order Uniformly Convergent Richardson Extrapolation Method for Singularly Perturbed Degenerate Parabolic PDEs

This article studies the numerical solution of singularly perturbed degenerate parabolic convection–diffusion problem on a rectangular domain. The solution of this problem exhibits a boundary layer in the neighborhood of left boundary of the domain. We discretize the domain by piecewise-uniform Shishkin mesh in the spatial direction and uniform mesh in the temporal direction. The numerical scheme consists of the implicit-Euler scheme for the time derivative and upwind finite difference scheme for the spatial derivatives. By applying the Richardson extrapolation technique, we improve the order of accuracy of the numerical solution from \(O (N^{-1} \ln ^2 N + \varDelta t)\) to \(O (N^{-2} \ln ^2 N + \varDelta t^2)\) (measured in the discrete supremum norm), where \(N+1\) mesh points are used in spatial direction and \(\varDelta t\) is the step size in the temporal direction. Error estimates are derived. Some numerical experiments are carried out to validate the theoretical results.

A finite volume method on general meshes for a degenerate parabolic convection–reaction–diffusion equation

We propose a finite volume method on general meshes for the discretization of a degenerate parabolic convection---reaction---diffusion equation. Equations of this type arise in many contexts, such as for example the modeling of contaminant transport in porous media. The diffusion term, which can be anisotropic and heterogeneous, is discretized using a recently developed hybrid mimetic mixed framework. We construct a family of discretizations for the convection term, which uses the hybrid interface unknowns. We consider a wide range of unstructured possibly nonmatching polyhedral meshes in arbitrary space dimension. The scheme is fully implicit in time, it is locally conservative and robust with respect to the Peclet number. We obtain a convergence result based upon a priori estimates and the Frechet---Kolmogorov compactness theorem. We implement the scheme both in two and three space dimensions and compare the numerical results obtained with the upwind and the centered discretizations of the convection term numerically.

Finite volume methods for degenerate chemotaxis model

A finite volume method for solving the degenerate chemotaxis model is presented, along with numerical examples. This model consists of a degenerate parabolic convection–diffusion PDE for the density of the cell-population coupled to a parabolic PDE for the chemoattractant concentration. It is shown that discrete solutions exist, and the scheme converges.

Discontinuous Galerkin relaxation algorithm for solute transport in porous media

AbstractWe consider a degenerate parabolic convection dominated equation which models the transport of contaminant in porous media. The numerical scheme is fulfilled by combining the DG – Discontinuous Galerkin Method with and an efficient relaxation algorithm that recently developed. Numerical results show the efficiency of our scheme. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems

We propose and analyze a numerical scheme for nonlinear degenerate parabolic convection–diffusion–reaction equations in two or three space dimensions. We discretize the diffusion term, which generally involves an inhomogeneous and anisotropic diffusion tensor, over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming (Crouzeix–Raviart) finite element method, or using the stiffness matrix of the hybridization of the lowest-order Raviart–Thomas mixed finite element method. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. Checking the local Peclet number, we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection-dominated case. This technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor. We prove the convergence of the scheme, only supposing the shape regularity condition for the original mesh. We use a priori estimates and the Kolmogorov relative compactness theorem for this purpose. The proposed scheme is robust, only 5-point (7-point in space dimension three), locally conservative, efficient, and stable, which is confirmed by numerical experiments.

On degenerate saturated-diffusion equations with convection

We study a class of degenerate parabolic convection–diffusion equations, endowed with a mechanism for saturation of the diffusion flux, which corrects the unphysical gradient-flux relations at high gradients. This paper extends our previous works on the effects of diffusion with saturation on convection and the impact of saturation on porous media-type diffusion, where it has been demonstrated that a nonlinear saturating diffusion is susceptible to a self-induced formation of discontinuities. In this work we demonstrate that nonlinear convection enhances the breakdown effect. We carry both analytical and numerical studies of the model equation, ut + f(u)x = [φ(u)Q(ux, u)]x, where Q is a bounded increasing function, φ(0) = 0 and φ(u) ∼ un, n > 0 for u ∼ 0. Depending on a choice of n, we obtain two distinctive processes. If 0 ≤ n ≤ 1, a discontinuity forms only when the upstream–downstream disparity exceeds a critical threshold, but if n > 1, all travelling waves are found to have a sharp discontinuous front. In fact, given a compact or a semi-compact initial datum, the front will not start to move until such a discontinuity forms.

ON A DIFFUSIVELY CORRECTED KINEMATIC-WAVE TRAFFIC FLOW MODEL WITH CHANGING ROAD SURFACE CONDITIONS

The well-known Lighthill–Whitham–Richards kinematic traffic flow model for unidirectional flow on a single-lane highway is extended to include both abruptly changing road surface conditions and drivers' reaction time and anticipation length. The result is a strongly degenerate convection–diffusion equation, where the diffusion term, accounting for the drivers' behavior, is effective only where the local car density exceeds a critical value, and the convective flux function depends discontinuously on the location. It is shown that the validity of the proposed traffic model is supported by a recent mathematical well-posedness (existence and uniqueness) theory for quasilinear degenerate parabolic convection–diffusion equations with discontinuous coefficients.20,22This theory includes a convergence proof for a monotone finite-difference scheme, which is used herein to simulate the traffic flow model for a variety of situations.