Convection-diffusion Problems Research Articles

An analysis of the bilinear finite volume method for the singularly-perturbed convection-diffusion problems on Shishkin mesh

In this paper, we study the bilinear finite volume element method for solving the singularly perturbed convection-diffusion problem on the Shishkin mesh. We first prove that the finite volume element scheme is ϵ-uniformly stable. Then, based on new expression of the finite volume bilinear form and some detailed integral calculations, an ϵ-uniform error estimation is derived in the ϵ-weighted gradient norm, including the L2-norm. This error estimate is better than the known result. Moreover, we also give the L∞-error estimate near the boundary layer regions. At last, numerical experiments show the effectiveness of our method.

Surrogate modeling of multi-dimensional premixed and non-premixed combustion using pseudo-time stepping physics-informed neural networks

Physics-informed neural networks (PINNs) have emerged as a promising alternative to conventional computational fluid dynamics (CFD) approaches for solving and modeling multi-dimensional flow fields. They offer instant inference speed and cost-effectiveness without the need for training datasets. However, compared to common data-driven methods, purely learning the physical constraints of partial differential equations and boundary conditions is much more challenging and prone to convergence issues leading to incorrect local optima. This training robustness issue significantly increases the difficulty of fine-tuning PINNs and limits their widespread adoption. In this work, we present improvements to the prior field-resolving surrogate modeling framework for combustion systems based on PINNs. First, inspired by the time-stepping schemes used in CFD numerical methods, we introduce a pseudo-time stepping loss aggregation algorithm to enhance the convergence robustness of the PINNs training process. This new pseudo-time stepping PINNs (PTS-PINNs) method is then tested in non-reactive convection–diffusion problem, and the results demonstrated its good convergence capability for multi-species transport problems. Second, the effectiveness of the PTS-PINNs method was verified in the case of methane–air premixed combustion, and the results show that the L2 norm relative error of all variables can be reduced within 5%. Finally, we also extend the capability of the PTS-PINNs method to address a more complex methane–air non-premixed combustion problem. The results indicate that the PTS-PINNs method can still achieve commendable accuracy by reducing the relative error to within 10%. Overall, the PTS-PINNs method demonstrates the ability to rapidly and accurately identify the convergence direction of the model, surpassing traditional PINNs methods in this regard.

Correction to: Implicit-Explicit Local Discontinuous Galerkin Methods with Generalized Alternating Numerical Fluxes for Convection-Diffusion Problems

Correction to: Implicit-Explicit Local Discontinuous Galerkin Methods with Generalized Alternating Numerical Fluxes for Convection-Diffusion Problems

Two-relaxation-time regularized lattice Boltzmann model for convection-diffusion equation with spatially dependent coefficients

In this paper, a new two-relaxation-time regularized (TRT-R) lattice Boltzmann (LB) model for the convection-diffusion equation (CDE) with spatially dependent coefficients is proposed. Within this framework, we first derive a TRT-R collision operator (CO) by constructing a new regularized procedure through the high-order Hermite expansion of non-equilibrium part. Then, a first-order discrete-velocity form of discrete source term is introduced to improve the accuracy of the source term. Finally, a new first-order space-derivative auxiliary term is proposed to recover the correct CDE. To assess this model, we simulated non-homogeneous convection-diffusion problems with adjustable diffusion intensity in both two and three dimensions. The findings indicate that the newly introduced source terms markedly enhance the model's precision and stability under varying time steps, grid resolutions, diffusion scaling coefficients, and magic parameters. The adoption of the TRT-R CO leads to a significant error reduction compared to the classic BGK CO in most scenarios. Furthermore, the influence of the magic parameter on the performance of the TRT-R CO was investigated. Beyond this, the study also confirms the efficacy of the TRT-R CO in eliminating numerical slip when enforcing Dirichlet boundary conditions with a halfway bounce-back scheme, thereby providing further evidence of the algorithm's advantages.

Construction of Second-Order Finite Difference Schemes for Diffusion-Convection Problems of Multifractional Suspensions in Coastal Marine Systems

Introduction. This paper addresses an initial-boundary value problem for the transport of multifractional suspensions applied to coastal marine systems. This problem describes the processes of transport, deposition of suspension particles, and the transitions between its various fractions. To obtain monotonic finite difference schemes for diffusion-convection problems of suspensions, it is advisable to use schemes that satisfy the maximum principle. When constructing a finite difference scheme that adheres to the maximum principle, it is desirable to achieve second-order spatial accuracy for bothinterior and boundary points of the domain under study. Materials and Methods. This problem presents certain difficulties when considering the boundaries of the geometric domain, where boundary conditions of the second and third kinds are applied. In these cases, to maintain second-order approximation accuracy, an “extended” grid is introduced (a grid supplemented with fictitious nodes). The guidelineis the approximation of the given boundary conditions using the central difference formula, with the exclusion of the concentration function at the fictitious node from the resulting expressions. Results. Second-order accurate finite difference schemes for the diffusion-convection problem of multifractional suspensions in coastal marine systems are constructed. Discussion and Conclusion. The proposed schemes are not absolutely stable, and a detailed analysis of stability and convergence, particularly concerning the grid step ratio, remains an important problem that the author plans to address in the future.

Reduced-dimensional modelling for nonlinear convection-dominated flow in cylindric domains

The aim of the paper is to construct and justify asymptotic approximations for solutions to quasilinear convection–diffusion problems with a predominance of nonlinear convective flow in a thin cylinder, where an inhomogeneous nonlinear Robin-type boundary condition involving convective and diffusive fluxes is imposed on the lateral surface. The limit problem for vanishing diffusion and the cylinder shrinking to an interval is a nonlinear first-order conservation law. For a time span that allows for a classical solution of this limit problem corresponding uniform pointwise and energy estimates are proven. They provide precise model error estimates with respect to the small parameter that controls the double viscosity-geometric limit. In addition, other problems with more higher Péclet numbers are also considered.

Streamline Diffusion Weak Galerkin Finite Element Methods for Linear Unsteady State Convection Diffusion Equations and Error Analysis

In this paper, the streamline diffusion weak Galerkin finite element method is proposed and analyzed for solving unsteady time convection diffusion problem in two dimension. The v-elliptic property and the stability of this scheme are proved in terms of some conditions. We derive an error estimate in L2(μ) and H1(μ) norm. Numerical experiments have demonstrated the effectiveness of the method in solving convection propagation problems, and the theoretical analysis has been validated.

Monotonic unbounded schemes transformer (MUST): an approach to remove undershoots and overshoots in family of unbounded schemes using finite-volume method

PurposeThis paper presents a Monotonic Unbounded Schemes Transformer (MUST) approach to bound/monotonize (remove undershoots and overshoots) unbounded spatial differencing schemes automatically, and naturally. Automatically means the approach (1) captures the critical cell Peclet number when an unbounded scheme starts to produce physically unrealistic solution automatically, and (2) removes the undershoots and overshoots as part of the formulation without requiring human interventions. Naturally implies, all the terms in the discretization equation of the unbounded spatial differencing scheme are retained.Design/methodology/approachThe authors do not formulate new higher-order scheme. MUST transforms an unbounded higher-order scheme into a bounded higher-order scheme.FindingsThe solutions obtained with MUST are identical to those without MUST when the cell Peclet number is smaller than the critical cell Peclet number. For cell Peclet numbers larger than the critical cell Peclet numbers, MUST sets the nodal values to the limiter value which can be derived for the problem at-hand. The authors propose a way to derive the limiter value. The authors tested MUST on the central differencing scheme, the second-order upwind differencing scheme and the QUICK differencing scheme. In all cases tested, MUST is able to (1) capture the critical cell Peclet numbers; the exact locations when overshoots and undershoots occur, and (2) limit the nodal value to the value of the limiter values. These are achieved by retaining all the discretization terms of the respective differencing schemes naturally and accomplished automatically as part of the discretization process. The authors demonstrated MUST using one-dimensional problems. Results for a two-dimensional convection–diffusion problem are shown in Appendix to show generality of MUST.Originality/valueThe authors present an original approach to convert any unbounded scheme to bounded scheme while retaining all the terms in the original discretization equation.

A comparative study on numerical methods for Fredholm integro-differential equations of convection-diffusion problem with integral boundary conditions

This paper numerically solves Fredholm integro-differential equations with small parameters and integral boundary conditions. The solution of these equations has a boundary layer at the right boundary. A central difference scheme approximates the second-order derivative, a backward difference (upwind scheme) approximates the first-order derivative, and the trapezoidal rule is used for the integral term with a Shishkin mesh. It is shown that theoretically, the proposed scheme is uniformly convergent with almost first-order convergence. Further to improve the order of convergence from first order to second order, we use the post-processing and the hybrid scheme. Two numerical examples are computed to support the theoretical results.

A hybrid interpolating element-free Galerkin method for 3D steady-state convection diffusion problems

This study investigates a hybrid interpolating element-free Galerkin (HIEFG) method for solving 3D convection diffusion problems. The HIEFG approach divides a 3D solution domain into a sequence of interconnected 2D sub-domains, and in these 2D sub-domains, the interpolating element-free Galerkin (IEFG) method is applied to form the discretized equations. The improved interpolating moving least-squares (IMLS) method is used to obtain the shape function of the IEFG method for 2D problems. The finite difference method is employed to combine the discretized equations in 2D sub-domains in the splitting direction. Then, the HIEFG method's formulas are derived for steady-state convection diffusion problems in 3D solution domain. Three numerical examples are used to discuss the impacts of the number of nodes, the number of split layers, and the scaling parameters of the influence domain on the computational precision and CPU time of the HIEFG technique. Imposing boundary conditions directly and the dimension splitting technique in this method significantly improves the computational speed greatly.

Uniform convergence of finite element method on Bakhvalov-type mesh for a 2-D singularly perturbed convection–diffusion problem with exponential layers

Analyzing uniform convergence of finite element method for a 2-D singularly perturbed convection–diffusion problem with exponential layers on Bakhvalov-type mesh remains a complex, unsolved problem. Previous attempts to address this issue have encountered significant obstacles, largely due to the constraints imposed by a specific mesh. These difficulties stem from three primary factors: the width of the mesh subdomain adjacent to the transition point, constraints imposed by the Dirichlet boundary condition, and the structural characteristics of exponential layers. In response to these challenges, this paper introduces a novel analysis technique that leverages the properties of interpolation and the relationship between the smooth function and the layer function on the boundary. By combining this technique with a simplified interpolation, we establish the uniform convergence of optimal order k under an energy norm for finite element method of any order k. Numerical experiments validate our theoretical findings.

On Loss Functionals for Physics-Informed Neural Networks for Steady-State Convection-Dominated Convection-Diffusion Problems

On Loss Functionals for Physics-Informed Neural Networks for Steady-State Convection-Dominated Convection-Diffusion Problems

A new local and explicit kinetic method for linear and non-linear convection-diffusion problems with finite kinetic speeds: II. Multi-dimensional case

We extend to multi-dimensions the work of [1], where new fully explicit kinetic methods were built for the approximation of linear and non-linear convection-diffusion problems. The fundamental principles from the earlier work are retained: (1) rather than aiming for the desired equations in the strict limit of a vanishing relaxation parameter, as is commonly done in the diffusion limit of kinetic methods, diffusion terms are sought as a first-order correction of this limit in a Chapman-Enskog expansion, (2) introducing a coupling between the conserved variables within the relaxation process by a specifically designed collision matrix makes it possible to systematically match a desired diffusion. Extending this strategy to multi-dimensions cannot, however, be achieved through simple directional splitting, as diffusion is likely to couple space directions with each other, such as with shear viscosity in the Navier-Stokes equations. In this work, we show how rewriting the collision matrix in terms of moments can address this issue, regardless of the number of kinetic waves, while ensuring conservation systematically. This rewriting allows for introducing a new class of kinetic models called regularized models, simplifying the numerical methods and establishing connections with Jin-Xin models. Subsequently, new explicit arbitrary high-order kinetic schemes are formulated and validated on standard two-dimensional cases from the literature. Excellent results are obtained in the simulation of a shock-boundary layer interaction, validating their ability to approximate the Navier-Stokes equations with kinetic speeds obeying nothing but a subcharacteristic condition along with a hyperbolic constraint on the time step.

Efficient finite element method for 2D singularly perturbed parabolic convection diffusion problems with discontinuous source term

Efficient finite element method for 2D singularly perturbed parabolic convection diffusion problems with discontinuous source term

PARAMETER-UNIFORM HYBRID NUMERICAL SCHEME FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION PROBLEMS WITH DISCONTINUOUS INITIAL CONDITIONS

This article presents a parameter-uniform hybrid numerical technique for singularly perturbed parabolic convection-diffusion problems (SPPCDP) with discontinuous initial conditions (DIC). It utilizes the classical backward-Euler technique for time discretization and a hybrid finite difference scheme ( which is a proper combination of the midpoint upwind scheme in the outer regions and the classical central difference scheme in the interior layer regions(generated by the DIC)) for spatial discretization. The scheme produces parameter-uniform numerical approximations on a piecewise- uniform Shishkin mesh. When the perturbation parameter ε(0 < ε ≪ 1) is small, it becomes difficult to solve these problems using the classical numerical methods (standard central difference or a standard upwind scheme) on uniform meshes because discontinuous initial conditions frequently appear in the solutions of this class of problems. The method is shown to converge uniformly in the discrete supremum with nearly second-order spatial accuracy. The suggested method is subjected to a stability study, and parameter-uniform error estimates are generated. In order to support the theoretical findings, numerical results are presented.

Numerical analysis of a class of penalty discontinuous Galerkin methods for nonlocal diffusion problems

In this paper, we consider a class of discontinuous Galerkin (DG) methods for one-dimensional nonlocal diffusion (ND) problems. The nonlocal models are integral equations that are widely used in describing many physical phenomena with long-range interactions. The ND problem is the nonlocal analog of the classic diffusion problem, and the nonlocality disappears as the interaction radius (horizon) vanishes, then the ND problem recovers the classic diffusion problem. Under certain conditions, the exact solution to the ND problem may contain discontinuities, and this is one of the properties of the ND problem that sets it apart from the classic diffusion problem. It is natural to adopt the DG method to compute the ND problems since the DG method shows its great advantages in resolving problems with discontinuities in the computational fluid dynamics in the past several decades. Based on \cite{DuCAMC2020}, we construct the DG methods with different penalty terms, and the proposed DG methods have their local counterparts as the horizon vanishes. This indicates the proposed methods will converge to the existing DG schemes as the horizon vanishes, which is crucial to obtain the {\it asymptotic compatibility}. Rigorous proofs are provided to demonstrate the stability, error estimates, and asymptotic compatibility of the proposed DG schemes. To see the nonlocal diffusion effect, we also consider the time-dependent convection-diffusion problems with nonlocal diffusion. We conduct several numerical experiments, including accuracy tests and Burgers' equation with nonlocal diffusion, and various horizons are taken to show the good performance of the proposed algorithm and validate the theoretical findings.

A new local and explicit kinetic method for linear and non-linear convection-diffusion problems with finite kinetic speeds: I. One-dimensional case

We propose a numerical approach, of the BGK kinetic type, that is able to approximate with a given, but arbitrary, order of accuracy the solution of linear and non-linear convection-diffusion type problems: scalar advection-diffusion, non-linear scalar problems of this type and the compressible Navier-Stokes equations. Our kinetic model can use finite advection speeds that are independent of the relaxation parameter, and the time step does not suffer from a parabolic constraint. Having finite speeds is in contrast with many of the previous works about this kind of approach, and we explain why this is possible: paraphrasing more or less [1], the convection-diffusion like PDE is not a limit of the BGK equation, but a correction of the same PDE without the parabolic term at the second order in the relaxation parameter that is interpreted as Knudsen number. We then show that introducing a relaxation matrix instead of the well-known BGK relaxation makes it possible to target a desired convection-diffusion system.Several numerical examples, ranging from a simple pure diffusion model to the compressible Navier-Stokes equations illustrate our approach.

Interpolated coefficients stabilizer-free weak Galerkin method for semilinear parabolic convection–diffusion problem

We continue our effort in Li et al. (2024) to explore an interpolated coefficients stabilizer-free weak Galerkin finite element method (IC SFWG-FEM) to solve a one-dimensional semilinear parabolic convection–diffusion equation. Due to the introduction of interpolated coefficients and the design without stabilizers, this method not only possesses the capability of approximating functions and sparsity in the stiffness matrix, but also reduces the complexity of analysis and programming. Theoretical analysis of stability for the semi-discrete IC SFWG finite element scheme is provided. Moreover, numerical experiments are carried out to demonstrate the effectivity and stability.

A parameter uniform numerical method on a Bakhvalov type mesh for singularly perturbed degenerate parabolic convection–diffusion problems

AbstractWe are focused on the numerical treatment of a singularly perturbed degenerate parabolic convection–diffusion problem that exhibits a parabolic boundary layer. The discretization and analysis of the problem are done in two steps. In the first step, we discretize in time and prove its uniform convergence using an auxiliary problem. In the second step, we discretize in space using an upwind scheme on a Bakhvalov-type mesh and prove its uniform convergence using the truncation error and barrier function approach, wherein several bounds derived for the mesh step sizes are used. Numerical results for a couple of examples are presented to support the theoretical bounds derived in the paper.

A modified cell-centered nodal integral scheme for the convection-diffusion equation

The nodal integral methods (NIMs) are very efficient and accurate coarse-mesh methods for solving partial differential equations. The cell-centered NIM (CCNIM) is a simplified variant of the NIMs that has recently shown its efficiency in solving fluid flow problems but has been hampered by issues such as inapplicability to one-dimensional problems, complexities in handling Neumann boundary conditions and the formulation of a system of differential-algebraic equations (DAEs) for discrete unknowns. Here, we present a modified version of the CCNIM designed to overcome the challenges associated with its previous version. Our novel development retains the essence of CCNIM while resolving these issues. The proposed scheme, grounded in the nodal framework, achieves second-order accuracy in both spatial and temporal dimensions. Unlike its precursor, the proposed method formulates algebraic equations for discrete variables per node, eliminating the cumbersome DAE system. Neumann boundary conditions are seamlessly incorporated through a straightforward flux definition, and applicability to one-dimensional problems is now feasible. We successfully apply our approach to one and two-dimensional convection-diffusion problems with known analytical solutions to validate our approach. The simplicity and robustness of the approach lay the foundation for its seamless extension to more complex fluid flow problems.