Convection-diffusion Problems Research Articles

An improved Kellogg-Tsan solution decomposition in numerical methods for singularly perturbed convection-diffusion problems

We consider the Kellogg-Tsan decomposition of the solution to the linear one-dimensional singularly perturbed convection-diffusion problem and improve it by including the solution of the corresponding reduced problem as a component. The upwind scheme on a modified Shishkin-type mesh is used to approximate the unknown component of the decomposition. It is proved that the error is O(ε(ln⁡ε)2N−1), where ε is the perturbation parameter and N is the number of mesh steps. The high accuracy of the method is illustrated by numerical examples.

Phase-field-lattice Boltzmann method for dendritic growth with melt flow and thermosolutal convection–diffusion

We propose a new phase-field model formulated within the system of lattice Boltzmann (LB) equation for simulating solidification and dendritic growth with fully coupled melt flow and thermosolutal convection–diffusion. With the evolution of the phase field and the transport phenomena all modeled and integrated within the same LB framework, this method preserves and combines the intrinsic advantages of the phase-field method (PFM) and the lattice Boltzmann method (LBM). Particularly, the present PFM/LBM model has several improved features compared to the existing phase-field models including: (1) a novel multiple-relaxation-time (MRT) LB scheme for the phase-field evolution is proposed to effectively model solidification coupled with melt flow and thermosolutal convection–diffusion with improved numerical stability and accuracy, (2) convenient diffuse interface treatments are implemented for the melt flow and thermosolutal transport which can be applied to the entire domain without tracking the interface, and (3) the evolution of the phase field, flow, concentration, and temperature fields on the level of microscopic distribution functions in the LB schemes is decoupled with a multiple-time-scaling strategy (despite their full physical coupling), thus solidification at high Lewis numbers (ratios of the liquid thermal to solutal diffusivities) can be conveniently modeled. The applicability and accuracy of the present PFM/LBM model are verified with four numerical tests including isothermal, iso-solutal and thermosolutal convection–diffusion problems, where excellent agreement in terms of phase-field and thermosolutal distributions and dendritic tip growth velocity and radius with those reported in the literature is demonstrated. The proposed PFM/LBM model can be an attractive and powerful tool for large-scale dendritic growth simulations given the high scalability of the LBM.

Finite element method on Shishkin mesh for a singularly perturbed problem with an interior layer

In this paper, we discuss a singularly perturbed convection diffusion problem, whose solution contains an interior layer caused by the discontinuity of the convection coefficient. For this kind of problem, uniform convergence of the finite element method has not been established. For this aim, after proving the minimum principle and stability result, we derive the asymptotic expansion of the solution and a priori estimates of its smooth and layer components. Then we construct a finite element method on Shishkin mesh, and prove uniform convergence of almost order k in the energy norm, where k is the order of piecewise polynomials.

Local and parallel finite element methods based on two‐grid discretizations for unsteady convection–diffusion problem

AbstractIn this article, some local and parallel finite element methods are proposed and investigated for the time‐dependent convection–diffusion problem. With backward Euler scheme for the temporal discretization, the basic idea of the present methods is that for a solution to the considered equations, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure at each time step. The partition of unity is used to collect the local high frequency components to assemble a global continuous approximation. Theoretical results are obtained and numerical tests are reported to support the theoretical findings.

An unfitted Eulerian finite element method for the time-dependent Stokes problem on moving domains

Abstract We analyse a Eulerian finite element method, combining a Eulerian time-stepping scheme applied to the time-dependent Stokes equations with the CutFEM approach using inf-sup stable Taylor–Hood elements for the spatial discretization. This is based on the method introduced by Lehrenfeld & Olshanskii (2019, A Eulerian finite element method for PDEs in time-dependent domains. ESAIM: M2AN, 53, 585–614) in the context of a scalar convection–diffusion problems on moving domains, and extended to the nonstationary Stokes problem on moving domains by Burman et al. (2019, arXiv:1910.03054 [math.NA]) using stabilized equal-order elements. The analysis includes the geometrical error made by integrating over approximated level set domains in the discrete CutFEM setting. The method is implemented and the theoretical results are illustrated using numerical examples.

Локально-двумерные схемы расщепления для параллельного решения трехмерной задачи транспорта взвешенного вещества

A 3D model of suspended matter transport in coastal marine systems is considered, which takes into account many factors, including the hydraulic size or the rate of particle deposition, the propagation of suspended matter, sedimentation, the intensity of distribution of suspended matter sources, etc. The difference operators of diffusion transport in the horizontal and vertical directions for this problem have significantly different characteristic spatiotemporal scales of processes, as well as spectra. With typical sampling, applied to shallow-water systems in the South of Russia (the Sea of Azov, the Tsimlyansk reservoir), the steps in horizontal directions are 200-1000 meters, the coefficients of turbulent exchange (turbulent diffusion) are (103-104) m2/sec; in the vertical direction - - - steps of 0.1 m-1 m, and the coefficients of microturbulent exchange in the vertical — (0.1-1) m2/sec. If we focus on the use of explicit locally twodimensional - - - locally one-dimensional splitting schemes, then the permissible values of the time step for a two-dimensional problem will be about 10-100 seconds, and for a one-dimensional problem in the vertical direction - - - 0.1 – 1 sec. This motivates us to construct an additive locally-two-dimensional-locallyonedimensional splitting scheme in geometric directions. The paper describes a parallel algorithm that uses both explicit and implicit schemes to approximate the two-dimensional diffusion-convection problem in horizontal directions and the one-dimensional diffusion-convection problem in the vertical direction. The two-dimensional implicit diffusion-convection problem in horizontal directions is numerically solved by the adaptive alternating-triangular method. The numerical implementation of the one-dimensional diffusion-convection problem in the vertical direction is carried out by a sequential run-through method for a series of independent one-dimensional three-point problems in the vertical direction on a given layer. To increase the efficiency of parallel calculations, the decomposition of the calculated spatial grid and all grid data in one or two spatial directions - in horizontal directions-is also performed. The obtained algorithms are compared taking into account the permissible values of time steps and the actual time spent on performing calculations and exchanging information on each time layer.

Second Order Monotone Finite-Difference Schemes on Non-Uniform Grids for Multi-Dimensional Convection-Diffusion Problem with a Boundary Condition of the Third Kind

Second Order Monotone Finite-Difference Schemes on Non-Uniform Grids for Multi-Dimensional Convection-Diffusion Problem with a Boundary Condition of the Third Kind

Performance of similarity explicit group iteration for solving 2D unsteady convection-diffusion equation

In this paper, a similarity finite difference (SFD) solution is addressed for thetwo-dimensional (2D) parabolic partial differential equation (PDE), specifically on the unsteady convection-diffusion problem. Structuring the similarity transformation using wave variables, we reduce the parabolic PDE into elliptic PDE. The numerical solution of the corresponding similarity equation is obtained using a second-order central SFD discretization schemeto get the second-order SFD approximation equation. We propose a four-point similarity explicit group (4-point SEG) iterative methodasa numericalsolution of the large-scale and sparse linear systems derived from SFD discretization of 2D unsteady convection-diffusion equation (CDE). To showthe 4-point SEG iteration efficiency, two iterative methods, such as Jacobiand Gauss-Seidel (GS) iterations, are also considered. The numerical experiments are carried out using three different problems to illustrate our proposed iterative method's performance. Finally, the numerical results showed that our proposed iterative method is more efficient than the Jacobiand GS iterations in terms of iteration number and execution time.

A Special Multigrid Strategy on Non-Uniform Grids for Solving 3D Convection–Diffusion Problems with Boundary/Interior Layers

Boundary or interior layer problems of high-dimensional convection–diffusion equations have distinct asymmetry. Consequently, computational grid distributions and linear algebraic systems arising from finite difference schemes for them are also asymmetric. Numerical solutions for these kinds of problems are more complicated than those symmetric problems. In this paper, we extended our previous work on the partial semi-coarsening multigrid method combined with the high-order compact (HOC) difference scheme for solving the two-dimensional (2D) convection–diffusion problems on non-uniform grids to the three-dimensional (3D) cases. The main merit of the present method is that the multigrid method on non-uniform grids can be performed with a different number of grids in different coordinate axes, which is more efficient than the multigrid method on non-uniform grids with the same number of grids in different coordinate axes. Numerical experiments are carried out to validate the accuracy and efficiency of the present method. It is shown that, without losing the high precision, the present method is very effective to reduce computing cost by cutting down the number of grids in the direction(s) which does/do not contain boundary or interior layer(s).

A diffusion–convection problem with a fractional derivative along the trajectory of motion

Abstract A new mathematical model of the diffusion–convective process with ‘memory along the flow path’ is proposed. This process is described by a homogeneous one-dimensional Dirichlet initial-boundary value problem with a fractional derivative along the characteristic curve of the convection operator. A finite-difference approximation of the problem is constructed and investigated. The stability estimates for finite-difference schemes are proved. The accuracy estimates are given for the case of sufficiently smooth input data and the solution.

Optimal Parameters for Third Order Runge–Kutta Exponential Integrators for Convection–Diffusion Problems

Optimal Parameters for Third Order Runge–Kutta Exponential Integrators for Convection–Diffusion Problems

Layer resolving fitted mesh method for parabolic convection-diffusion problems with a variable diffusion

Layer resolving fitted mesh method for parabolic convection-diffusion problems with a variable diffusion

Local maximum entropy approximation-based streamline upwind Petrov–Galerkin meshfree method for convection–diffusion problem

Local maximum entropy approximation-based streamline upwind Petrov–Galerkin meshfree method for convection–diffusion problem

Numerical approximations to a singularly perturbed convection-diffusion problem with a discontinuous initial condition

A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified which matches the discontinuity in the initial condition and also satisfies the homogenous parabolic differential equation associated with the problem. The difference between this analytical function and the solution of the parabolic problem is approximated numerically, using an upwind finite difference operator combined with an appropriate layer-adapted mesh. The numerical method is shown to be parameter-uniform. Numerical results are presented to illustrate the theoretical error bounds established in the paper.

The Darcy problem with porosity depending exponentially on the pressure

We consider the flow of a viscous incompressible fluid through a porous medium. We allow the permeability of the medium to depend exponentially on the pressure and provide an analysis for this model. We study a splitting formulation where a convection diffusion problem is used to define the permeability, which is then used in a linear Darcy equation. We also study a discretization of this problem, and provide an error analysis for it.

An LT-BEM for an unsteady diffusion-convection problem of another class of anisotropic FGMs

An LT-BEM is used to find numerical solutions to an unsteady variable coefficients diffusion-convection equation governing problems of another class of anisotropic functionally graded media. A boundary-only integral equation is derived, following a transformation of the variable coefficients equation to a constant coefficients equation. The boundary-only integral equation is used to find the numerical solutions. Some problems considered are those for several new types of functionally graded materials of compressible or incompressible flow. The effects of the anisotropy and inhomogeneity of the considered materials on the solutions are also shown as to say that it is very important to take the anisotropy and inhomogeneity into account in experimental or practical studies for any applications.

An enriched Galerkin-characteristics finite element method for convection-dominated and transport problems

We propose an enriched Galerkin-characteristics finite element method for numerical solution of convection-dominated problems. The method uses the modified method of characteristics for the integration of the total derivative in time, combined with the finite element method for the spatial discretization on unstructured grids. The L2-projection method is implemented for the evaluation of numerical solutions by tracking the departure points from the integration points in each element. In the present study, a family of quadrature rules are used to enrich the approximation of integrals in the L2-projection method. The use of quadrature rules as an enrichment procedure allows for spatial discretizations on coarse fixed meshes and no need to introduce time-dependent enrichments. This procedure offers a very great advantage over the conventional Galerkin-characteristics finite element method since the same governing matrix representation can be used during the entire time steeping process. We also propose a multilevel adaptive procedure in the enriched Galerkin-characteristics finite element method by monitoring the gradient of the solution in the computational domain during its advection. In comparison with the traditional finite element analysis with h-, p- and hp-version refinements, the present approach is much simpler, more robust and efficient, and it yields more accurate solutions for a fixed number of degrees of freedom without refining the mesh. To examine the performance of the proposed method we solve several test examples for convection-diffusion problems. Comparison to the conventional Galerkin-characteristics finite element method is also carried out in the present work. The aim of such enriched method compared to the classical finite element method is to solve the time-dependent convection-dominated problems efficiently and with an appropriate level of accuracy.

The condition for the conservation of momentum at the interface under phase transitions of solutions

The present work considers a change in the momentum under the transfer of a solution through the interface. It is shown that pressure related to the partial volumes of the components arises as a result of the kinetics of the addition of particles to a new phase in the solution. In contrast to the description of one-dimensional interphase mass transfer using the convective diffusion problem, the proposed model provides the stationary exponential distributions of components in both phases. It is shown that generalized Fick’s law provides the condition for the conservation of the momentum of a component at the interface. The model describes component segregation by the interface.

Solving the diffusion-convection problem using MPI parallel computing technology

The aim of the work is to build a software package for the distributed solution of the problem of matter transfer in a reservoir. A parallel implementation of the grid domain decomposition methods for computationally complex diffusion-convection problems, taking into account the architecture and parameters of a multiprocessor computing system located on the basis of the infrastructure facility of the University of Science and Technology «Sirius», is considered. An algorithm for parallel solution of the formulated hydrophysics problem using MPI technology has been developed. The analysis of the efficiency of the developed parallel algorithms is carried out and the study of the obtained solutions of computationally complex problems of aquatic ecology is carried out.

The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Diffusion Problems

In this paper, a fully discrete stability analysis is carried out for the direct discontinuous Galerkin (DDG) methods coupled with Runge-Kutta-type implicit-explicit time marching, for solving one-dimensional linear convection-diffusion problems. In the spatial discretization, both the original DDG methods and the refined DDG methods with interface corrections are considered. In the time discretization, the convection term is treated explicitly and the diffusion term implicitly. By the energy method, we show that the corresponding fully discrete schemes are unconditionally stable, in the sense that the time-step $$\tau$$ is only required to be upper bounded by a constant which is independent of the mesh size h. Optimal error estimate is also obtained by the aid of a special global projection. Numerical experiments are given to verify the stability and accuracy of the proposed schemes.