Convection-dominated Diffusion Equations Research Articles

Numerical discretization of a Darcy–Forchheimer flow with variable density and heat transfer

In this paper, we study a heat transfer scenario in Darcy–Forchheimer porous media with variable density. The block-centered finite difference method is applied to discretize the non-isothermal flow equations governing the system. Specifically, the pressure field is modeled using the nonlinear Darcy–Forchheimer formulation, while the density and temperature are described by convection-dominated diffusion equations, which are treated via the characteristic method. Theoretical analyses are rigorously developed for pressure, velocity, density, temperature, and auxiliary flux across non-uniform grids. Several numerical experiments are carried out to illustrate the merits of our method by comparing numerical results to analytical solutions.

Maximum-Principle-Preserving High-Order Conservative Difference Schemes for Convection-Dominated Diffusion Equations

Maximum-Principle-Preserving High-Order Conservative Difference Schemes for Convection-Dominated Diffusion Equations

An upwind-mixed volume element method on changing meshes for compressible miscible displacement problem

Numerical simulation of oil-water miscible displacement is discussed in this paper, and a compressible problem of energy mathematics is solved potentially. The mathematical model defined by a nonlinear system includes mainly two partial differential equations (PDEs): a parabolic equation for the pressure and a convection-diffusion equation for the saturation. The pressure is obtained by a conservative mixed finite volume element method (MFVE). The computational accuracy is improved for Darcy velocity. A conservative upwind mixed finite volume element method (UMFVE) is applied to compute the saturation on changing meshes. The diffusion is discretized by a mixed finite volume element method, and the convection is computed by upwind differences. The upwind method can solve convection-dominated diffusion equations accurately and avoids numerical dispersion and nonphysical oscillation. The saturation and the adjoint vector function are obtained simultaneously. An optimal order error estimates is obtained. Finally, numerical examples are provided to show the accuracy, efficiency and possible applications.

A Stable Mimetic Finite-Difference Method for Convection-Dominated Diffusion Equations

A Stable Mimetic Finite-Difference Method for Convection-Dominated Diffusion Equations

Unconditional superconvergence analysis of modified finite difference streamlined diffusion method for nonlinear convection-dominated diffusion equation

In this article, a Crank-Nicolson fully discrete finite element scheme of modified finite difference streamlined diffusion (MFDSD) method is developed and investigated for nonlinear convection-dominated diffusion equation, which can get rid of the numerical oscillation appeared in Galerkin finite element method (FEM). The supercloseness and superconvergence estimates of order O(h2+τ2) in H1 norm are derived without the restriction between the time step τ and the mesh size h. Firstly, a time discrete system is established to split the error into two parts - the temporal error and spatial error, and the regularity of the solution of the time discrete system is deduced with the help of mathematical induction. Then the numerical solution is bounded in L∞ norm by the spatial error which leads to the above unconditional supercloseness property, and the global superconvergence result is deduced through interpolation post-processing technique. Lastly, two numerical examples are provided to verify the correctness of the theoretical analysis and to show the big advantage of the proposed MFDSD method over the Galerkin FEM.

Positivity preserving temporal second-order spatial fourth-order conservative characteristic methods for convection dominated diffusion equations

Positivity preserving temporal second-order spatial fourth-order conservative characteristic methods for convection dominated diffusion equations

Convergence analysis of weak Galerkin finite element method for semilinear parabolic convection dominated diffusion equations on polygonal meshes

Convergence analysis of weak Galerkin finite element method for semilinear parabolic convection dominated diffusion equations on polygonal meshes

Uniformly convergent scheme for steady MHD duct flow problems with high Hartmann numbers on structured and unstructured grids

The steady incompressible Magnetohydrodynamic (MHD) duct flow problems with a transverse magnetic field at high Hartmann numbers (Ha) is a coupled system of convection-dominated diffusion equations and the solutions contain sharp boundary layers. We develop a five-point scheme and an edge-centered scheme on structured and unstructured grids with tailored finite point method (TFPM) for solving the coupled system of convection–diffusion equations. The five-point scheme is an upwind difference scheme, where the coefficients of the difference operator are tailored to some particular properties of the convection–diffusion equation with constant coefficients on the local cell. The edge-centered scheme is constructed by the continuity of the normal fluxes for each edge. The normal flux is expressed in terms of the edge-centered unknowns defined at the local cell and the unknowns at the adjacent cells are not needed. Uniform second order convergence can be obtained by the edge-centered scheme on structured and unstructured grids over a wide range of Ha 10 to 106. Both of the two schemes can correctly reproduce the fast change of the solutions which contain sharp boundary layers at high Ha even without refining the mesh. Numerical examples are provided to show the accuracy and of the TFPM.

A reduced-order characteristic finite element method based on POD for optimal control problem governed by convection–diffusion equation

This paper proposes a reduced-order characteristic finite element (ROCFE) approximation with several unknowns for optimal control problem governed by convection–diffusion equation. It inherits the advantages of the characteristic methods in solving convection-dominated diffusion equations. For this problem, the usual snapshots extraction way cannot be used, so we propose a new snapshots extraction way. Then the proper orthogonal decomposition (POD) is used on unsteady state and co-state systems, which efficiently reduces the number of unknowns and computational costs. Optimal a priori error estimates are derived for the state, co-state and control. Some numerical experiments are carried out to show the accuracy and efficiency of the ROCFE method.

Positive and Conservative Characteristic Block-Centered Finite Difference Methods for Convection Dominated Diffusion Equations

Positive and Conservative Characteristic Block-Centered Finite Difference Methods for Convection Dominated Diffusion Equations

SUPG-Stabilized Virtual Element Method for Optimal Control Problem Governed by a Convection Dominated Diffusion Equation.

In this paper, the streamline upwind/Petrov Galerkin (SUPG) stabilized virtual element method (VEM) for optimal control problem governed by a convection dominated diffusion equation is investigated. The virtual element discrete scheme is constructed based on the first-optimize-then-discretize strategy and SUPG stabilized virtual element approximation of the state equation and adjoint state equation. An a priori error estimate is derived for both the state, adjoint state, and the control. Numerical experiments are carried out to illustrate the theoretical findings.

Novel finite point approach for solving time-fractional convection-dominated diffusion equations

In this paper, a stabilized numerical method with high accuracy is proposed to solve time-fractional singularly perturbed convection-diffusion equation with variable coefficients. The tailored finite point method (TFPM) is adopted to discrete equation in the spatial direction, while the time direction is discreted by the G-L approximation and the L1 approximation. It can effectively eliminate non-physical oscillation or excessive numerical dispersion caused by convection dominant. The stability of the scheme is verified by theoretical analysis. Finally, one-dimensional and two-dimensional numerical examples are presented to verify the efficiency of the method.

Fourth‐order compact scheme based on quasi‐variable mesh for three‐dimensional mildly nonlinear stationary convection–diffusion equations

AbstractA new family of compact schemes of increased accuracy using quasi‐variable mesh is presented for determining approximate solutions to the three‐space dimensions mildly nonlinear convection dominated diffusion equations. The main thought behind the proposed scheme is to get uniformly distributed local truncation error, which otherwise not possible in case of finite‐difference discretization using constant step‐sizes mesh points. According to the zero or nonzero values of mesh stretching quantities, the increased accuracy fourth‐order method refers to uniform meshes or quasi‐variable meshes schemes. It is easy to tune the mesh points depending upon the location of subdomains having comparatively more fluctuating solution behavior. The block‐tridiagonal construction of the Jacobian (iteration matrix) acquired with the new difference scheme makes it easier to implement with a reasonable computing time. We will describe the matrix and graph theoretic approach to analyze the convergence property and error estimate of the generalized scheme. A measure of accuracy, such as maximum errors, root‐mean‐squared errors, and numerical convergence rate, is examined by solving various forms of convection‐dominated diffusion equations.

Stability Analysis of Quasi-variable Grids Cubic Spline Fourth-Order Compact Implicit Algorithms for Burger’s Type Parabolic PDEs

In the present paper, we describe cubic spline combined with high accuracy compact formulation for computing approximate solution values to nonlinear convection-dominated diffusion equations, and apply the method to several variants of Burger’s type parabolic partial differential equations. A combination of cubic spline interpolating polynomial and quasi-variable grids sequence yields a two-level implicit compact computational scheme that falls in the range of increased accuracy. The grid stretching parameter plays an important role while considering the boundary layer phenomenon to convection-dominated parabolic equations. The discretization on the quasi-variable grid network yields a more accurate and precise solution than those obtained on a uniform grid network. It is easy to extend the proposed scheme to the singular equation with compact operators, preserving the order and accuracy. It is shown through stability analysis that the new method is stable with an accuracy of order four and two in the spatial and temporal direction, respectively. Numerical illustrations with generalized Burgers–Fisher equation, Burgers–Huxley equation, and some physically relevant parabolic partial differential equations corroborate the theoretical analysis.

A nearly-conservative high-order Lagrange–Galerkin method for the resolution of scalar convection-dominated equations in non-divergence-free velocity fields

In this work, we present a novel Lagrange–Galerkin method for the resolution of the scalar pure-convection and convection-dominated diffusion equations in non-divergence-free velocity fields. The scheme has been formulated so that (i) time-integration is carried out by means of an arbitrary order backward differentiation formula, (ii) finite element space discretizations of any order can be employed, and (iii) it satisfies a discrete mass balance equation at each simulation instant and preserves mass in those cases in which no fluid enters nor leaves the control domain, as long as the integrals involved are computed exactly. These properties are achieved by posing a conservation law for a weighted mass –from which the standard mass can be seen as a particular case– that can be easily discretized in time and in space with any order of accuracy. Numerical experiments –which include a three-dimensional test– show that the method is as accurate in time as the employed time-marching formula, as accurate in space as the finite element discretization, and generally more accurate –especially in terms of mass conservation errors– and efficient than the classic, non-conservative Lagrange–Galerkin method. To the best of our knowledge, this is the first conservative Lagrange–Galerkin method for convection-dominated equations that considers both non-divergence-free velocity fields and time and space discretizations of any order.

A Mixed Volume Element With Characteristic Mixed Volume Element for Contamination Transport Problem

Nonlinear systems of convection-dominated diffusion equations are used as the mathematical model of contamination transport problem which is an important topic in environ mental protection science. An elliptic equation defines the pressure, a convection-diffusion equation expresses the concentration of contamination, and an ordinary differential equation interprets the surface absorption concentration. The transport pressure appears in the equation of the concentration which determines the Darcy velocity and also controls the physical process. The method of conservative mixed volume element is used to solve the flow equation which improves the computational accuracy of Darcy velocity by one order. We use the mixed volume element with the characteristic to approximate the concentration. This method of characteristic not only preserves the strong computational stability at sharp front, but also eliminates numerical dispersion and nonphysical oscillation. In the present scheme, we could adopt a large step without losing accuracy. The diffusion is approximated by the mixed volume element. The concentration and its adjoint vector function are obtained simultaneously, and the locally conservative law is preserved. An optimal second order estimates in l2-norm is derived.

Tailored finite point method for time fractional convection dominated diffusion problems with boundary layers

We propose a tailored finite point method (TFPM) for solving time fractional convection dominated diffusion equations in this paper. The main idea of TFPM is to firstly approximate the diffusion, convection coefficient near each grid by a constant, and then determine the weights of the finite difference scheme by using the exact solution of the convection diffusion equation with constant coefficients. This adaptation perfectly captures the rapid transition of the solutions which contain sharp boundary layers even with coarse meshes. The accuracy and stability of the scheme are rigorously analyzed. Numerical examples are shown to verify the accuracy and reliability of the proposed scheme.

Modified weak Galerkin method with weakly imposed boundary condition for convection-dominated diffusion equations

In this paper, a modified weak Galerkin (MWG) finite element method with weakly imposed boundary conditions is presented for solving convection-dominated diffusion equations. The method is shown uniformly stable for all diffusion parameters. The method converges at the optimal order for large diffusion problems in the energy norm, and at half a super-convergent order for small diffusion problems. Various numerical examples are presented, showing that the method is as effective as the weak Galerkin method.

Mixed volume element with characteristic mixed volume element method for compressible contamination treatment from nuclear waste

This paper discusses numerical simulation of three-dimensional compressible contamination treatment from nuclear waste in porous media. The mathematical model includes five major unknowns, the pressure, Darcy velocity, brine, radionuclide and the temperature, determined by a parabolic equation, convection-dominated diffusion equations and the heat conductor equation. Considering the physical properties and the characters of equations, a mixed volume element with the characteristics is presented. For the flow equation, a conservative mixed volume element is used to obtain the pressure and Darcy velocity, and improves the accuracy of the latter. The characteristics could solve the convection dominated problems well and confirm the stability and accuracy at the front sharps, thus a characteristic mixed volume element is given for computing the concentrations and temperature. The composite procedures have the nature of conservation. Convergence analysis is discussed. Finally, an experimental system of partial differential equations is carried out and the feasibility and computational efficiency are illustrated. This method could solve such complicated problems well.

An adaptive strategy for solving convection dominated diffusion equation

In this work, an adaptive element free Galerkin (EFG) technique is proposed for solving convection diffusion equation. A post-processed gradient is used as a posteriori error estimation to find locations with large contribution of the error. Also, to avoid instabilities, the EFG method is applied on a modified equation instead of the original equation. In the modified equation diffusion coefficient (known as artificial diffusion) depends to the distance between nodal points. The numerical results reveal efficiency of the adaptive technique.