Convection-dominated Diffusion Research Articles

An Oscillation-Free Discontinuous Galerkin Method for a Nonlinear Stochastic Convection-Dominated Diffusion Problem and Its Error Analysis

An Oscillation-Free Discontinuous Galerkin Method for a Nonlinear Stochastic Convection-Dominated Diffusion Problem and Its Error Analysis

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

Super-localized orthogonal decomposition for convection-dominated diffusion problems

This paper presents a novel multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The method involves applying the solution operator to piecewise constant right-hand sides on an arbitrary coarse mesh, which defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-norm, the Galerkin projection onto this generalized finite element space even yields ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}-independent error bounds, ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}-robust convergence without pre-asymptotic effects even in the under-resolved regime of large mesh Péclet numbers.

Error Analysis of the Classical Artificial Diffusion Weak Galerkin Finite Element Method for the steady state- convection diffusion-reaction Equation in 2-D

In this study, we modified the error in the weak Galerkin method when solving problems in which diffusion is the dominant convection( h) in two dimensions. This is done by adding the artificial diffusion term (-δΔw, where δ=h-ϵ). The finite element method for discrete functions using a weakly defined gradient operator is presented in this study. The concept of the weak discrete gradient is introduced, which plays an important role when using numerical methods to solve partial differential equations. The goal of this study is to enhance the accuracy and stability of the solutions by studying the ellipticity and stability properties of the method, which works to ensure that the numerical method retains the properties of the original equation while reducing the fluctuations occurring with the weak galerkin finite element method. Specific theories have been used to estimate the error in parameters -norm, and . Practical examples demonstrate how this method improves the handling of partial equations characterized by convection-dominated diffusion, enhancing its potential for advancing numerical simulations in engineering and physics.

A novel local meshless collocation method with partial upwind scheme for solving convection-dominated diffusion problems

A novel local meshless collocation method with partial upwind scheme for solving convection-dominated diffusion problems

Solving Minimal Residual Methods in W-1,p′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{-1,p'}$$\\end{document} with Large Exponents p

We introduce a numerical scheme that approximates solutions to linear PDE’s by minimizing a residual in the W-1,p′(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{-1,p'}(\\Omega )$$\\end{document} norm with exponents p>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p> 2$$\\end{document}. The resulting problem is solved by regularized Kačanov iterations, allowing to compute the solution to the non-linear minimization problem even for large exponents p≫2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\gg 2$$\\end{document}. Such large exponents remedy instabilities of finite element methods for problems like convection-dominated diffusion.

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.

A variational multiscale method derived from an adaptive stabilized conforming finite element method via residual minimization on dual norms

This paper interprets the stabilized finite element method via residual minimization as a variational multiscale method. We approximate the solution to the partial differential equations using two discrete spaces that we build on a triangulation of the domain; we denote these spaces as coarse and enriched spaces. Building on the adaptive stabilized finite element method via residual minimization, we find a coarse-scale approximation in a continuous space by minimizing the residual on a dual discontinuous Galerkin norm; this process allows us to compute a robust error estimate to construct an on-the-fly adaptive method. We reinterpret the residual projection using the variational multiscale framework to derive a fine-scale approximation. As a result, on each mesh of the adaptive process, we obtain stable coarse- and fine-scale solutions derived from a symmetric saddle-point formulation and an a-posteriori error indicator to guide automatic adaptivity. We test our framework in several challenging scenarios for linear and nonlinear convection-dominated diffusion problems to demonstrate the framework’s performance in providing stability in the solution with optimal convergence rates in the asymptotic regime and robust performance in the pre-asymptotic regime. Lastly, we introduce a heuristic dual-term contribution in the variational form to improve the full-scale approximation for symmetric formulations (e.g., diffusion problem).

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.

NIPG Finite Element Method for Convection-Dominated Diffusion Problems with Discontinuous Data

This paper presents the nonsymmetric interior penalty Galerkin (NIPG) finite element method for a class of one-dimensional convection dominated diffusion problems with discontinuous coefficients. The solution of the considered class of problem exhibits boundary and interior layers. Piecewise uniform Shishkin-type meshes are used for the spatial discretization. The error estimates in the energy norm have been derived for the proposed schemes. Theoretical results are supported by conducting numerical experiments. It is established that the errors are uniform with respect to the perturbation parameter [Formula: see text]. The uniformness of the error estimates with the perturbation parameter [Formula: see text] has also been established numerically for [Formula: see text]- norm.

A high-resolution fuzzy transform combined compact scheme for 2D nonlinear elliptic partial differential equations

This paper proposes a new high-resolution fuzzy transform algorithm for solving two-dimensional nonlinear elliptic partial differential equations (PDEs). The underlying new computational method implements the method of so-called approximating fuzzy components, which evaluate the solution values with fourth-order accuracy at internal mesh points. Triangular basic functions and fuzzy components are locally determined by linear combinations of solution values at nine points. Such a scheme connects the proposed method of approximating fuzzy components with the exact values of the solution using a linear system of equations. Compact approximations of high-resolution fuzzy components using nine points give a block tridiagonal Jacobi matrix. Apart from the numerical solution, it is easy to construct closed-form approximate solutions using a 2D spline interpolation polynomial from the available data with fuzzy components. The upper bounds of the approximation errors are estimated, as well as the convergence of the approximating solutions. Simulations with linear and nonlinear elliptical PDEs arising from quantum mechanics and convection-dominated diffusion phenomena are presented to confirm the usefulness of the new scheme and fourth-order convergence. To summarize:•The paper presents a high-resolution numerical method for the two-dimensions elliptic PDEs with nonlinear terms.•The combined effect of fuzzy transform and compact discretizations yields almost fourth-order accuracies to Schro¨dinger equation, convection-diffusion equation, and Burgers equation.•The high-order numerical scheme is computationally efficient and employs minimal data storage.

A new GPU-based corrected explicit-implicit domain decomposition scheme for convection-dominated diffusion problems

In this study, a class of GPU-based corrected explicit-implicit domain decomposition schemes (GCEIDD) is proposed to accelerate the solution of convection-dominated diffusion problems on GPUs. All of these methods take advantage of the fractional steps and corrected explicit-implicit domain decomposition techniques. In each sweep, the domain is decomposed into many strip-divided subdomains. The aim is to reduce the size and increase the number of tri-diagonal systems to provide enough parallelism to keep the GPU occupied. In different steps of the algorithms, we use low-complexity schemes with a high degree of parallelism. The generic form of GCEIDD allows different schemes for the prediction, correction, and one-dimensional implicit solution. We have proposed two methods based on modified upwind and characteristics finite difference and a combined method that bypasses the limitations of these two. Also, important aspects of CUDA programming, which are critical to the implementation of the GCEIDD algorithm, are discussed in detail. For solving tri-diagonal systems, a three-stage strategy is proposed, which creates a balance between occupancy, cache hit rate, and shared memory usage and reduces cache contentions. Results show that GCEIDD has good accuracy and stability even when the number of subdomains is very large. The proposed method can accelerate the solution by a factor of up to 3 compared with the GPU implementation of the classic fractional step methods.

A higher-order uniformly convergent defect correction method for singularly perturbed convection-diffusion problems on an adaptive mesh

This paper presents a defect correction scheme based on finite-difference discretization for a singularly perturbed convection-dominated diffusion problem. The solution of this class of problems exhibits a multiscale character. There are narrow regions in which the solution grows exponentially and reveals layer behaviour. The defect correction method that we propose improves the efficiency of a numerical solution through iterative improvement and generates a stable higher-order method over an adaptive non-uniform polynomial-Shishkin mesh. An extensive theoretical analysis is presented, which establishes that the method is second-order uniformly convergent and highly stable. The convergence obtained is optimal because it is free from any logarithmic term. The numerical result for two model problems is presented, agreeing with the theoretical estimates. Furthermore, we compare the results with those of other non-uniform mesh found in the literature.

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.

A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems

We present a Petrov-Galerkin (PG) method for a class of nonlocal convection-dominated diffusion problems. There are two main ingredients in our approach. First, we define the norm on the test space as induced by the trial space norm, i.e., the optimal test norm, so that the inf-sup condition can be satisfied uniformly independent of the problem. We show the well-posedness of a class of nonlocal convection-dominated diffusion problems under the optimal test norm with general assumptions on the nonlocal diffusion and convection kernels. Second, following the framework of Cohen et al. (2012), we embed the original nonlocal convection-dominated diffusion problem into a larger mixed problem so as to choose an enriched test space as a stabilization of the numerical algorithm. In the numerical experiments, we use an approximate optimal test norm which can be efficiently implemented in 1d, and study its performance against the energy norm on the test space. We conduct convergence studies for the nonlocal problem using uniform h- and p-refinements, and adaptive h-refinements on both smooth manufactured solutions and solutions with sharp gradient in a transition layer. In addition, we confirm that the PG method is asymptotically compatible, following the notion in Tian and Du (2014).

Variational multiscale virtual element method for the convection-dominated diffusion problem

In this paper a new stabilized virtual element method for the convection-dominated diffusion problem is proposed. By taking advantage of robustness of variational multiscale method and flexibility of virtual element method, the proposed method is proven to be very easy to implement. The adding stabilized term in the variational formulation can be calculated directly using the stability portion of the virtual element method. There is no user-defined parameter. And the priori error estimate can be easily obtained as there is no change in the consistency portion in the virtual element method. Numerical experiments show good agreement with the theoretical convergence rate and validate the stabilization effect.

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.