Robust numerical scheme for singularly perturbed convection–diffusion parabolic initial–boundary-value problems on equidistributed grids

In this article, we study the numerical solution of singularly perturbed parabolic convection–diffusion problems exhibiting regular boundary layers. To solve these problems, we use the classical upwind finite difference scheme on layer-adapted nonuniform meshes. The nonuniform meshes are obtained by equidistributing a positive monitor function, which depends on the second-order spatial derivative of the singular component of the solution. The truncation error and the stability analysis are obtained. Parameter-uniform error estimates are derived for the numerical solution. Semilinear IBVPs are also solved. Numerical experiments are carried out to support the theoretical results.     

Analysis of lumped approximations in the finite-element method for convection–diffusion problems

The Petrov–Galerkin finite-element method with a lumped mass matrix is analyzed. It is stated that in some cases it excessively smoothes the solutions and causes large errors. It is shown that weight functions can be chosen, which eliminate the above-mentioned drawbacks. The corresponding approximations are constructed in the form of systems of ordinary differential equations and finite-difference schemes. The theoretical results are confirmed by calculated data.     

High-order compact boundary value method for the solution of unsteady convection–diffusion problems

In this paper, we propose a new class of high-order accurate methods for solving the two-dimensional unsteady convection–diffusion equation. These techniques are based on the method of lines approach. We apply a compact finite difference approximation of fourth order for discretizing spatial derivatives and a boundary value method of fourth order for the time integration of the resulted linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. Also this method is unconditionally stable due to the favorable stability property of boundary value methods. Numerical results obtained from solving several problems include problems encounter in many transport phenomena, problems with Gaussian pulse initial condition and problems with sharp discontinuity near the boundary, show that the compact finite difference approximation of fourth order and a boundary value method of fourth order give an efficient algorithm for solving such problems.     

Crank–Nicolson finite difference method based on a midpoint upwind scheme on a non-uniform mesh for time-dependent singularly perturbed convection–diffusion equations

A numerical approach is proposed to examine the singularly perturbed time-dependent convection–diffusion equation in one space dimension on a rectangular domain. The solution of the considered problem exhibits a boundary layer on the right side of the domain. We semi-discretize the continuous problem by means of the Crank–Nicolson finite difference method in the temporal direction. The semi-discretization yields a set of ordinary differential equations and the resulting set of ordinary differential equations is discretized by using a midpoint upwind finite difference scheme on a non-uniform mesh of Shishkin type. The resulting finite difference method is shown to be almost second-order accurate in a coarse mesh and almost first-order accurate in a fine mesh in the spatial direction. The accuracy achieved in the temporal direction is almost second order. An extensive amount of analysis has been carried out in order to prove the uniform convergence of the method. Finally we have found that the resulting method is uniformly convergent with respect to the singular perturbation parameter, i.e. ?-uniform. Some numerical experiments have been carried out to validate the proposed theoretical results.     

Relaxation method for unsteady convection–diffusion equations

We propose and implement a relaxation method for solving unsteady linear and nonlinear convection–diffusion equations with continuous or discontinuity-like initial conditions. The method transforms a convection–diffusion equation into a relaxation system, which contains a stiff source term. The resulting relaxation system is then solved by a third-order accurate implicit–explicit (IMEX) Runge–Kutta method in time and a fifth-order finite difference WENO scheme in space. Numerical results show that the method can be used to effectively solve convection–diffusion equations with both smooth structures and discontinuities.     

A parameter-free dynamic diffusion method for advection–diffusion–reaction problems

In this paper, we present a two-scale finite element formulation, named Dynamic Diffusion (DD), for advection–diffusion–reaction problems. By decomposing the velocity field in coarse and subgrid scales, the latter is used to determine the smallest amount of artificial diffusion to minimize the coarse-scale kinetic energy. This is done locally and dynamically, by imposing some constraints on the resolved scale solution, yielding a parameter-free consistent method. The subgrid scale space is defined by using bubble functions, whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. Convergence tests on a two-dimensional example are reported, yielding optimal rates. In addition, numerical experiments show that DD method is robust for a wide scope of application problems.     

Higher-order convergence with fractional-step method for singularly perturbed 2D parabolic convection–diffusion problems on Shishkin mesh

In this article, we propose a second-order uniformly convergent numerical method for a singularly perturbed 2D parabolic convection–diffusion initial–boundary-value problem. First, we use a fractional-step method to discretize the time derivative of the continuous problem on uniform mesh in the temporal direction, which gives a set of two 1D problems. Then, we use the classical finite difference scheme to discretize those 1D problems on a special mesh, which results almost first-order convergence, i.e., $O(N^{?1+\beta }lnN+\Delta t)$. To enhance the order of convergence to $O(N^{?2+\beta }{ln}^{2}N+\Delta t^2)$, we use the Richardson extrapolation technique. In support of the theoretical results, numerical experiments are performed by employing the proposed technique.     

Uniform supercloseness of Galerkin finite element method for convection–diffusion problems with characteristic layers

In this paper, we consider a singularly perturbed convection–diffusion equation posed on the unit square, where the solution has two characteristic layers and an exponential layer. A Galerkin finite element method on a Shishkin mesh is used to solve this problem. Its bilinear forms in different subdomains are carefully analyzed by means of a series of integral inequalities; a delicate analysis for the characteristic layers is needed. Based on these estimations, we prove supercloseness bounds of order $3∕2$ (up to a logarithmic factor) on triangular meshes and of order $2$ (up to a logarithmic factor) on hybrid meshes respectively. The result implies that the hybrid mesh, which replaces the triangles of the Shishkin mesh by rectangles in the exponential layer region, is superior to the Shishkin triangular mesh. Numerical experiments illustrate these theoretical results.     

Parameterized ABD preconditioning technique for time-periodic convection–diffusion problems

In this paper, a parameterized additive block diagonal (PABD) preconditioning technique is present for solving the nine-point approximations of the time-periodic convection–diffusion problems. The explicit expressions for the eigenvalues and eigenvectors of the corresponding preconditioned matrices are presented. The range of the optimal parameters is derived. Numerical experiments show that the generalized minimal residual method preconditioned by the PABD preconditioner with the experimental optimal parameters or some special values is effective for a wide range of problem sizes.     

Local projection stabilisation on S-type meshes for convection–diffusion problems with characteristic layers

Singularly perturbed convection–diffusion problems with exponential and characteristic layers are considered on the unit square. The discretisation is based on layer-adapted meshes. The standard Galerkin method and the local projection scheme are analysed for bilinear and higher order finite element where enriched spaces were used. For bilinears, first order convergence in the ε-weighted energy norm is shown for both the Galerkin and the stabilised scheme. However, supercloseness results of second order hold for the Galerkin method in the ε-weighted energy norm and for the local projection scheme in the corresponding norm. For the enriched ${\mathcal{Q}_p}$ -elements, p ≥ 2, which already contain the space ${\mathcal{P}_{p+1}}$ , a convergence order p + 1 in the ε-weighted energy norm is proved for both the Galerkin method and the local projection scheme. Furthermore, the local projection methods provides a supercloseness result of order p + 1 in local projection norm.     

A stabilizing augmented grid for rectangular discretizations of the convection–diffusion–reaction problems

We propose a numerical method for approximate solution of the convection–diffusion–reaction problems in the case of small diffusion. The method is based on the standard Galerkin finite element method on an extended space defined on the original grid plus a subgrid, where the original grid consists of rectangular elements. On each rectangular elements, we construct a subgrid with few points whose locations are critical for the stabilization of the problem, therefore they are chosen specially depending on some specific conditions that depend on the problem data. The resulting subgrid is combined with the initial coarse mesh, eventually, to solve the problem in the framework of Galerkin method on the augmented grid. The results of the numerical experiments confirm that the proposed method shows similar stability features with the well-known stabilized methods for the critical range of problem parameters.     

A CCD–ADI method for unsteady convection–diffusion equations

With a combined compact difference scheme for the spatial discretization and the Crank–Nicolson scheme for the temporal discretization, respectively, a high-order alternating direction implicit method (ADI) is proposed for solving unsteady two dimensional convection–diffusion equations. The method is sixth-order accurate in space and second-order accurate in time. The resulting matrix at each ADI computation step corresponds to a triple-tridiagonal system which can be effectively solved with a considerable saving in computing time. In practice, Richardson extrapolation is exploited to increase the temporal accuracy. The unconditional stability is proved by means of Fourier analysis for two dimensional convection–diffusion problems with periodic boundary conditions. Numerical experiments are conducted to demonstrate the efficiency of the proposed method. Moreover, the present method preserves the higher order accuracy for convection-dominated problems.     

Richardson extrapolation technique for singularly perturbed parabolic convection–diffusion problems

This paper deals with the study of a post-processing technique for one-dimensional singularly perturbed parabolic convection–diffusion problems exhibiting a regular boundary layer. For discretizing the time derivative, we use the classical backward-Euler method and for the spatial discretization the simple upwind scheme is used on a piecewise-uniform Shishkin mesh. We show that the use of Richardson extrapolation technique improves the ε-uniform accuracy of simple upwinding in the discrete supremum norm from O (N ⁻¹ ln N + Δt) to O (N ⁻² ln² N + Δt ²), where N is the number of mesh-intervals in the spatial direction and Δt is the step size in the temporal direction. The theoretical result is also verified computationally by applying the proposed technique on two test examples.     

Variational principles for advection–diffusion problems

Variational principles for linear and semilinear advection–diffusion problems with velocity field given by potential flow are described and analyzed. Mixed Dirichlet and prescribed flux conditions are treated. Existence and uniqueness results are proved and equivalent integral operator equations are found. A positive multiplier function related to the potential of the flow is used to change the system to divergence form. The dependence of the solution on inhomogeneous flux boundary data is determined.     

A split least-squares characteristic mixed element method for nonlinear nonstationary convection–diffusion problem

In this paper, a split least-squares characteristic mixed finite element method is proposed for solving nonlinear nonstationary convection–diffusion problem. By selecting the least-squares functional property, the resulting least-squares procedure can be split into two independent symmetric positive definite sub-schemes. The first sub-scheme is for the unknown variable u, which is the same as the standard characteristic Galerkin finite element approximation. The second sub-scheme is for the unknown flux σ. Theoretical analysis shows that the method yields the approximate solutions with optimal accuracy in L ²(Ω) norm for the primal unknown and in H(div; Ω) norm for the unknown flux, respectively. Some numerical examples are given to confirm our theory results.     

A coupled Newton iterative mixed finite element method for stationary conduction–convection problems

In this paper, a coupled Newton iterative mixed finite element method (MFEM) for solving the stationary conduction–convection problems in two dimension is given. In our method, the Newton iterative MFEM is used for solving all the equations of the conduction–convection problems. The stability and convergence analysis in H ¹-norm of ${u_h^n, T_h^n}$ and the L ²-norm of ${p_h^n}$ are derived. The theory analysis shows that our method is stable and have a good precision. Some numerical results are also given, which show that the coupled Newton iterative MFEM is highly efficient for the stationary conduction–convection problems.     

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

In 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 ?-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.     

Development of a sixth-order two-dimensional convection–diffusion scheme via Cole–Hopf transformation

In this paper, we develop a two-dimensional finite-difference scheme for solving the time-dependent convection–diffusion equation. The numerical method exploits Cole–Hopf equation to transform the nonlinear scalar transport equation into the linear heat conduction equation. Within the semi-discretization context, the time derivative term in the transformed parabolic equation is approximated by a second-order accurate time-stepping scheme, resulting in an inhomogeneous Helmholtz equation. We apply the alternating direction implicit scheme of Polezhaev to solve the Helmholtz equation. As the key to success in the present simulation, we develop a Helmholtz scheme with sixth-order spatial accuracy. As is standard practice, we validated the code against test problems which were amenable to exact solutions. Results show excellent agreement for the one-dimensional test problems and good agreement with the analytical solution for the two-dimensional problem.