Finite volume discretisation for the one-dimensional convection diffusion-dissipation equation (original) (raw)

Numerical modelling of convection–diffusion–adsorption problems in 1D using dynamical discretization

Two candidates for optimal numerical methods for highly nonlinear partial differential equations are compared. The chosen methods are characterized by their speed, accuracy and simplicity of formulation. They can be easily formulated in high level scripting languages like MATLAB or python and are suited for practical implementation. The setting is that of 1D general convection-diffusionadsorption-reaction systems, a setting of high relevance in chemical and groundwater engineering. First, the mathematical model is numerically solved by the method of lines (MOL) using a space discretization with moving grid points (r-adaptivity). The advantage is that with minimal grid points one captures the sharp fronts of the solution which can arise due to a strong adsorption. A large variety of isotherms can be included in the adsorption model for both equilibrium and non-equilibrium modes. In the second method, the mathematical model is approximated using interface modelling. However, this method is only applicable for adsorption in equilibrium mode. The numerical efficiency of the methods is discussed and the obtained numerical results are compared to determine their optimal use. Both methods are very suitable for solving inverse problems in practical implementations, as they are robust and fast. (J. Kačur), bm@cage.UGent.be (B. Malengier), trojakova@fmph.uniba.sk (E. Trojakov a).

The Method of Lines with a Finite Volumes Approach for Transient Convection-Diffusion Problems

Journal of Mathematical Sciences: Advances and Applications, 2017

This paper deals with development and analysis of a discretization by the method of lines with a finite volumes approach for a one-dimensional convection-diffusion equation. This discretization leads to an ordinary differential equation (ODE). We use an explicit scheme to solve the obtained ODE. ∞ L stability of the approximate solution under appropriate CFL conditions is also obtained. Study is done for constant and discontinuous thermal conductivities. Results of numerical experiments using the present approach are reported.

Discretization methods with analytical characteristic methods for advection-diffusion-reaction equations and 2d applications

ESAIM: Mathematical Modelling and Numerical Analysis, 2009

Our studies are motivated by a desire to model long-time simulations of possible scenarios for a waste disposal. Numerical methods are developed for solving the arising systems of convectiondiffusion-dispersion-reaction equations, and the received results of several discretization methods are presented. We concentrate on linear reaction systems, which can be solved analytically. In the numerical methods, we use large time-steps to achieve long simulation times of about 10 000 years. We propose higher-order discretization methods, which allow us to use large time-steps without losing accuracy. By decoupling of a multi-physical and multi-dimensional equation, simpler physical and one-dimensional equations are obtained and can be discretized with higher-order methods. The results can then be coupled with an operator-splitting method. We discuss benchmark problems given in the literature and real-life applications. We simulate a radioactive waste disposals with underlying flowing groundwater. The transport and reaction simulations for the decay chains are presented in 2d realistic domains, and we discuss the received results. Finally, we present our conclusions and ideas for further works.

Numerical Methods for Solving Convection-Diffusion Problems

2012

Convection-diffusion equations provide the basis for describing heat and mass transfer phenomena as well as processes of continuum mechanics. To handle flows in porous media, the fundamental issue is to model correctly the convective transport of individual phases. Moreover, for compressible media, the pressure equation itself is just a time-dependent convection-diffusion equation. For different problems, a convection-diffusion equation may be be written in various forms. The most popular formulation of convective transport employs the divergent (conservative) form. In some cases, the nondivergent (characteristic) form seems to be preferable. The so-called skew-symmetric form of convective transport operators that is the half-sum of the operators in the divergent and nondivergent forms is of great interest in some applications. Here we discuss the basic classes of discretization in space: finite difference schemes on rectangular grids, approximations on general polyhedra (the finite volume method), and finite element procedures. The key properties of discrete operators are studied for convective and diffusive transport. We emphasize the problems of constructing approximations for convection and diffusion operators that satisfy the maximum principle at the discrete level-they are called monotone approximations. Two-and three-level schemes are investigated for transient problems. Unconditionally stable explicit-implicit schemes are developed for convection-diffusion problems. Stability conditions are obtained both in finite-dimensional Hilbert spaces and in Banach spaces depending on the form in which the convection

An evaluation of eight discretization schemes for two-dimensional convection-diffusion equations

International Journal for Numerical Methods in Fluids, 1986

A comparative study of eight discretization schemes for the equations describing convection-diffusion transport phenomena is presented. The (differencing) schemes considered are the conventional central, upwind and hybrid difference together with the quadratic ~p s t r e a m ,~,~ quadratic upstream extended4 and quadratic upstream extended revised difference4 schemes. Also tested are the so called locally exact difference scheme5 and the power difference scheme.6 In multi-dimensional problems errors arise from 'false diffusion' and function approximations. It is asserted that false diffusion is essentially a multi-dimensional source of error. Hence errors associated with false diffusion may be investigated only via two-and three-dimensional problems. The above schemes have been tested for both one-and two-dimensional flows with sources, to distinguish between 'discretization' errors and 'false diffusion' err01-s.~ The one-dimensional study is reported in Reference 7. For 2D flows, the quadratic upstream difference schemes are shown to be superior in accuracy to the others at all Peclet numbers, for the test cases considered. The stability of the schemes and their CPU time requirements are also discussed.

An assessment of discretizations for convection-dominated convection–diffusion equations

Computer Methods in Applied Mechanics and Engineering, 2011

The performance of several numerical schemes for discretizing convection-dominated convection-diffusion equations will be investigated with respect to accuracy and efficiency. Accuracy is considered in measures which are of interest in applications. The study includes an exponentially fitted finite volume scheme, the Streamline-Upwind Petrov-Galerkin (SUPG) finite element method, a spurious oscillations at layers diminishing (SOLD) finite element method, a finite element method with continuous interior penalty (CIP) stabilization, a discontinuous Galerkin (DG) finite element method, and a total variation diminishing finite element method (FEMTVD). A detailed assessment of the schemes based on the Hemker example will be presented.

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

Numerische Mathematik, 2013

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 the modeling of contaminant transport in porous media. We discretize the diffusion term, which can be anisotropic and heterogeneous, via a hybrid finite volume scheme. We construct a partially upwind scheme for the convection term. We consider a wide range of unstructured possibly non-matching polygonal meshes in arbitrary space dimension. The only assumption on the mesh is that the volume elements must be star-shaped. The scheme is fully implicit in time, it is locally conservative and robust with respect to the Péclet number. We obtain a convergence result based upon a priori estimates and the Fréchet-Kolmogorov compactness theorem.

The Discrete Duality Finite Volume Method for Convection-diffusion Problems

SIAM Journal on Numerical Analysis, 2010

In this paper we extend the Discrete Duality Finite Volume (DDFV) formulation to the steady convection-diffusion equation. The discrete gradients defined in DDFV are used to define a cellbased gradient for the control volumes of both the primal and dual meshes, in order to achieve a higher-order accurate numerical flux for the convection term. A priori analysis is carried out to show convergence of the approximation and a global first-order convergence rate is derived. The theoretical results are confirmed by some numerical experiments.