Nonlinear Convection Diffusion Equation Research Articles

Linearization technique and its application to numerical solution of bidimensional nonlinear convection diffusion equation

Linearization technique and its application to numerical solution of bidimensional nonlinear convection diffusion equation

A new family of (5,5)CC-4OC schemes applicable for unsteady Navier–Stokes equations

In this paper, a new family of implicit compact finite difference schemes for computation of unsteady convection–diffusion equation with variable convection coefficient is proposed. The schemes which are fourth order accurate in space and second or lower order accurate in time depending on the choice of weighted time average parameter are then applied to unsteady Navier–Stokes system. The proposed schemes, where transport variable and its first derivatives are carried as the unknowns, combine virtues of compact discretization and Padé scheme for spatial derivative. These schemes which are based on a five point stencil with constant coefficients, named as “(5,5) Constant Coefficient 4th Order Compact” [(5,5)CC-4OC], give rise to a diagonally dominant system of equations and shows higher accuracy and better phase and amplitude error characteristics than some of the standard methods. These schemes are capable of using a grid aspect ratio other than unity and are unconditionally stable. They efficiently capture both transient and steady solutions of linear and nonlinear convection–diffusion equations with Dirichlet as well as Neumann boundary conditions. Subsequently the proposed schemes are applied to problems governed by the incompressible Navier–Stokes equations. The results obtained are in excellent agreement with analytical and available numerical results in all cases, establishing efficiency and accuracy of the proposed schemes.

A hybrid mixed method for the compressible Navier–Stokes equations

We present a novel discretization method for nonlinear convection–diffusion equations and, in particular, for the compressible Navier–Stokes equations. The method is based on a Discontinuous Galerkin (DG) discretization for convection terms, and a Mixed method using H (div) spaces for the diffusive terms. Furthermore, hybridization is used to reduce the number of globally coupled degrees of freedom. For the scalar case, a local postprocessing procedure is used to enhance the quality of the approximate solution wh. The method reduces to a DG scheme for pure convection, and to a Mixed method for pure diffusion, while for the intermediate case the combined variational formulation requires no additional parameters. We formulate and validate our scheme for nonlinear model problems, as well as compressible flow problems.

The Arithmetic Mean Solver in Lagged Diffusivity Method for Nonlinear Diffusion Equations

Th is paper deals with the solution of nonlinear system arising fro m finite difference discretization of nonlinear diffusion convection equations by the lagged diffusivity functional iteration method co mbined with d ifferent inner iterative solvers. The analysis of the whole procedure with the splitt ing methods of the Arith met ic Mean (AM) and of the Alternating Group Exp licit (A GE) has been developed. A comparison in terms of number of iterations has been done with the BiCG-STA B algorith m. So me nu merical experiments have been carried out and they seem to show the effectiveness of the lagged diffusivity procedure with the Arithmet ic Mean method as inner solver.

Exact solutions to the nonlinear diffusion-convection equation with variable coefficients and source term

The nonlinear diffusion-convection equation f(x)ut=(g(x)D(u)ux)x+h(x)P(u)ux+q(x)Q(u) with variable coefficients and source term has been studied. This equation is symmetrically reduced by the generalized conditional symmetry method. Some exact solutions to the resulting equations are constructed, with the diffusion terms D(u)=um (m≠-1,0,1) and D(u)=eu. These exact solutions are also the generalized functional separable solutions. Solutions to the equation with constant coefficients are covered by those exact solutions to the equation with variable coefficients.

Regularized Nonlinear Solvers for IMEX Methods Applied to Diffusively Corrected Multispecies Kinematic Flow Models

Implicit-explicit (IMEX) methods are suitable for the solution of nonlinear convection-diffusion equations, since the stability restrictions, coming from the explicitly treated convective part, are much less severe than those that would be deduced from an explicit treatment of the diffusive term. These schemes usually combine an explicit Runge--Kutta scheme for the time integration of the convective part with a diagonally implicit one for the diffusive part. The application of these schemes to multispecies kinematic flow models with strongly degenerate diffusive corrections requires the solution of highly nonlinear and nonsmooth systems of algebraic equations. Since the efficient solution of these systems by the Newton--Raphson method requires some degree of smoothness, it is proposed to regularize the diffusion coefficients in the model and to apply suitable techniques to solve these nonlinear systems in an efficient way. Numerical examples arising from models of polydisperse sedimentation and multiclass traffic flow confirm the efficiency of the methods proposed.

Optimal spatial error estimates for DG time discretizations

In this paper a general parabolic problem is considered and discretized by the discontinuous Galerkin (DG) method in time and, in general, in space. Optimal a priori error estimates in space as well as in time are derived and applied to the heat equation and to a nonlinear convection-diffusion equation.

Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection–diffusion equations on triangular meshes

We propose second order accurate discontinuous Galerkin (DG) schemes which satisfy a strict maximum principle for general nonlinear convection–diffusion equations on unstructured triangular meshes. Motivated by genuinely high order maximum-principle-satisfying DG schemes for hyperbolic conservation laws (Perthame, 1996) and (Zhang, 2010) [14,26], we prove that under suitable time step restriction for forward Euler time stepping, for general nonlinear convection–diffusion equations, the same scaling limiter coupled with second order DG methods preserves the physical bounds indicated by the initial condition while maintaining uniform second order accuracy. Similar to the purely convection cases, the limiters are mass conservative and easy to implement. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle. Following the idea in Zhang (2012) [30], we extend the schemes to two-dimensional convection–diffusion equations on triangular meshes. There are no geometric constraints on the mesh such as angle acuteness. Numerical results including incompressible Navier–Stokes equations are presented to validate and demonstrate the effectiveness of the numerical methods.

The Dirichlet-to-Neumann map for a nonlinear diffusion–convection equation

A method for constructing the Dirichlet-to-Neumann map for a nonlinear diffusion–convection equation is presented. The problem is reduced to the solution of a nonlinear integral equation in one independent variable. Existence and uniqueness of the solution may be proven for small times via a contraction mapping technique.

Classification and approximate solutions to perturbed diffusion–convection equations

Approximate symmetries of the perturbed nonlinear diffusion–convection equations are completely classified by the method originated with Fushchich and Shtelen. Moreover, for some interesting cases, symmetry reductions and approximate solutions are discussed in detail.

An Inverse Problem for a Nonlinear Diffusion-Convection Equation

A method for constructing the Dirichlet-to-Neumann map for a nonlinear diffusion-convection equation is presented. The problem is reduced to the solution of a nonlinear integral equation in one independent variable. Existence and uniqueness of the solution may be proven for small times via a contraction mapping technique.

Homogenization of a stochastic nonlinear reaction–diffusion equation with a large reaction term: The almost periodic framework

Homogenization of a stochastic nonlinear reaction–diffusion equation with a large nonlinear term is considered. Under a general Besicovitch almost periodicity assumption on the coefficients of the equation we prove that the sequence of solutions of the said problem converges in probability towards the solution of a rather different type of equation, namely, the stochastic nonlinear convection–diffusion equation which we explicitly derive in terms of appropriate functionals. We study some particular cases such as the periodic framework, and many others. This is achieved under a suitable generalized concept of Σ-convergence for stochastic processes.

IIM-based ADI finite difference scheme for nonlinear convection–diffusion equations with interfaces

Based on Li’s immersed interface method (IIM), an ADI-type finite difference scheme is proposed for solving two-dimensional nonlinear convection–diffusion interface problems on a fixed cartesian grid, which is unconditionally stable and converges with two-order accuracy in both time and space in maximum norm. Correction terms are added to the right-hand side of standard ADI scheme at irregular points. The nonlinear convection terms are treated by Adams–Bashforth method, without affecting the stability of difference schemes. A new method for computing the correction terms is developed, in which the Adams–Bashforth method is employed. Thus we can get an explicit approximation for the computation of corrections, when the jump condition is solution-dependent. Three numerical experiments are displayed and analyzed. The numerical results show good agreement with the exact solutions and confirm the convergence order.

Lagged diffusivity method for the solution of nonlinear diffusion convection problems with finite differences

This article concerns with the analysis of an iterative procedure for the solution of a nonlinear nonstationary diffusion convection equation in a two-dimensional bounded domain supplemented by Dirichlet boundary conditions. This procedure, denoted Lagged Diffusivity method, computes the solution by lagging the diffusion term. A model problem is considered and a finite difference discretization for that model is described. Furthermore, properties of the finite difference operator are proved. Then, a sufficient condition for the convergence of the Lagged Diffusivity method is given. At each stage of the iterative procedure, a linear system has to be solved and the Arithmetic Mean method is used. Numerical experiments show the efficiency, for different test functions, of the Lagged Diffusivity method combined with the Arithmetic Mean method as inner solver. Better results are obtained when the convection term increases.

Time Implicit High-Order Discontinuous Galerkin Method with Reduced Evaluation Cost

An efficient and robust time integration procedure for a high-order discontinuous Galerkin method is introduced for solving nonlinear second-order partial differential equations. The time discretization is based on an explicit formulation for the hyperbolic term and an implicit formulation for the parabolic term. The procedure uses an iterative algorithm with reduced evaluation cost. The size of the linear system to be solved is greatly reduced thanks to partial uncoupling in space between low-order and high-order degrees of freedom. Numerical examples are presented for the nonlinear convection-diffusion equation in one and two dimensions including steady and unsteady flow problems. The performance of the present method is investigated in terms of CPU time and compared to a fully implicit method. A von Neumann stability analysis is carried out in order to determine the stability and damping properties of the method. Besides a fairly reduced CPU effort, numerical results demonstrate better convergence properties of the present algorithm when compared to the fully implicit method.

On a non-linear moving boundary problem for a diffusion–convection equation

We study a one-dimensional free boundary problem for a non-linear diffusion–convection equation whose diffusivity is heterogeneous in space as well as being non-linear. Under the Bäcklund transformation the problem is reduced to an associated free boundary problem. We prove the existence and uniqueness, local in time, of the solution by using the Friedman Rubinstein integral representation method and the Banach contraction theorem.

Modified lattice Boltzmann scheme for nonlinear convection diffusion equations

To further improve the simulation performance of nonlinear convection diffusion equations (NCDE), in this paper, a modified lattice Boltzmann (LB) scheme is put forward by introducing a new parameter that offers more opportunities in studying NCDE while increases the computation cost hardly, the derivation of LB scheme is investigated in detail, and the numerical experiments are carried out in different NCDE, including Nonlinear Heat Conduction Equation, Burgers–Fisher Equation and Nonlinear Schrödinger Equation. The simulation results obtained not only agree well with the analytical solutions, but also demonstrate that the proposed scheme under the best choice of the parameters can preserve the better stability and higher accuracy in comparison with previous schemes.

Multidomain pseudospectral methods for nonlinear convection-diffusion equations

Multidomain pseudospectral approximations to nonlinear convection-diffusion equations are considered. The schemes are formulated with the Legendre-Galerkin method, but the nonlinear term is collocated at the Legendre/Chebyshev-Gauss-Lobatto points inside each subinterval. Appropriate base functions are introduced so that the matrix of the system is sparse, and the method can be implemented efficiently and in parallel. The stability and the optimal rate of convergence of the methods are proved. Numerical results are given for both the single domain and the multidomain methods to make a comparison.

Travelling waves in a convection–diffusion equation

We study existence and stability of travelling waves for nonlinear convection–diffusion equations in the 1-D Euclidean space. The diffusion coefficient depends on the gradient in analogy with the p-Laplacian and may be degenerate. Unconditional stability is established with respect to initial data perturbations in L 1 ( R ) .

Rectangular lattice Boltzmann model for nonlinear convection–diffusion equations

In this paper, a rectangular lattice Boltzmann model is proposed for nonlinear convection-diffusion equations (NCDEs). The model can be used to solve NCDEs with very general form by using a real/complex-valued quadric equilibrium distribution function and relaxation time. Detailed simulations on several examples are performed to validate the model. The numerical results show good agreement with the analytical solutions, and the numerical accuracy is much better than that of the models with a linear equilibrium distribution function.