Nonlinear Convection-diffusion Problems Research Articles

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.

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.

A relaxation approach to modeling properties of hyperbolic–parabolic type models

In this work, we propose a novel relaxation modeling approach for partial differential equations (PDEs) involving convective and diffusive terms. We reformulate the original convection–diffusion problem as a system of hyperbolic equations coupled with relaxation terms. In contrast to existing literature on relaxation modeling, where the solution of the reformulated problem converges to certain types of equations in the diffusive limit, our formalism treats the augmented problem as a system of coupled hyperbolic equations with relaxation acting on both the convective flux and the source term. Furthermore, we demonstrate that the new system of equations satisfies Liu’s sub-characteristic condition. To verify the robustness of our proposed approach, we perform numerical experiments on various important models, including nonlinear convection–diffusion problems with discontinuous coefficients. The results show the promising potential of our relaxation modeling approach for both pure and applied mathematical sciences, with applications in different models and areas.

Решение уравнений конвекции-диффузии локальным разрывным методом Галеркина

In the work, the discontinuous Galerkin method (LDG) is applied to solving linear and nonlinear convection-diffusion problems. The idea of this method is to transform high-order equations into a system of first-order equations, and then apply the classical discontinuous Galerkin method (DG) to this system. An orthogonal system of Legendre polynomials is chosen as the system of basis functions. The cases of both continuous and discontinuous solutions of the problem are considered. The dependence of the accuracy and stability of the method on the step of the spatial grid and the number of basis functions is investigated. The application of parallel computing to the algorithm is investigated. In the future, it is proposed to apply the LRG method for solving the problems of modeling gas mixtures flows.

Unified Integral Transforms and Non-Classical Eigenvalue Problems in Heat and Mass Transfer

Abstract The generalized integral transform technique (GITT) is reviewed as a computational–analytical methodology in linear and nonlinear convection–diffusion problems, based on eigenfunction expansions extracted from characteristic differential operators, coefficients, and boundary conditions present in the original partial differential problem formulation. Here, the emphasis is on the employment of nonclassical eigenvalue problems as the expansion basis, which do not fall into the more usual framework of Sturm–Liouville problems. The goal is to enable or improve the eigenfunction expansions convergence, by incorporating more information from the original operators into the chosen eigenvalue problem, while requiring the handling of such a more involved expansion base. In this concern, the proposed differential eigenvalue problem can itself be handled by the GITT, leading to an algebraic eigensystem analysis. Different classes of nonclassical eigenvalue problems are then reviewed and associated with typical applications in heat and mass transfer. Representative test cases are then chosen to illustrate the extended methodology and demonstrate the convergence rates attainable by this enhanced hybrid solution path.

A fast method for solving time-dependent nonlinear convection diffusion problems

<abstract><p>In this paper, a fast scheme for solving unsteady nonlinear convection diffusion problems is proposed and analyzed. At each step, we firstly isolate a nonlinear convection subproblem and a linear diffusion subproblem from the original problem by utilizing operator splitting. By Taylor expansion, we explicitly transform the nonlinear convection one into a linear problem with artificial inflow boundary conditions associated with the nonlinear flux. Then a multistep technique is provided to relax the possible stability requirement, which is due to the explicit processing of the convection problem. Since the self-adjointness and coerciveness of diffusion subproblems, there are so many preconditioned iterative solvers to get them solved with high efficiency at each time step. When using the finite element method to discretize all the resulting subproblems, the major stiffness matrices are same at each step, that is the reason why the unsteady nonlinear systems can be computed extremely fast with the present method. Finally, in order to validate the effectiveness of the present scheme, several numerical examples including the Burgers type and Buckley-Leverett type equations, are chosen as the numerical study.</p></abstract>

Gaussian radial basis functions method for linear and nonlinear convection–diffusion models in physical phenomena

Abstract In this study, we propose a simple direct meshless scheme based on the Gaussian radial basis function for the one-dimensional linear and nonlinear convection–diffusion problems, which frequently occur in physical phenomena. This is fulfilled by constructing a simple ‘anisotropic’ space–time Gaussian radial basis function. According to the proposed scheme, there is no need to remove time-dependent variables during the whole solution process, which leads it to a really meshless method. The suggested meshless method is implemented to the challenging convection–diffusion problems in a direct way with ease. Numerical results show that the proposed meshless method is simple, accurate, stable, easy-to-program and efficient for both linear and nonlinear convection–diffusion equation with different values of Péclet number. To assess the accuracy absolute error, average absolute error and root-mean-square error are used.

A high-order modified finite-volume method on Cartesian grids for nonlinear convection–diffusion problems

Recently, Buchmuller and Helzel proposed a modified dimension-by-dimension finite-volume (FV) WENO method on Cartesian grids for multidimensional nonlinear conservation laws which can retain the full order of accuracy of the underlying one-dimensional (1D) reconstruction. In this work, we extend this method to multidimensional convection–diffusion equations. The 1D sixth-order central reconstruction of the conserved quantity is utilized for discretizing the diffusion terms in which the diffusion coefficients may be nonlinear functions of the conserved quantity. Using high-order accurate conversions between edge-averaged values and edge center values of any sufficiently smooth quantity, high-order accurate convective and viscous numerical fluxes at cell interfaces are computed. The present modified FV method uses fourth-order accurate conversions for the diffusive fluxes. Numerical examples show that the present method achieves fourth-order accuracy for multidimensional smooth problems, and is suitable for the numerical simulation of viscous shocked flows.

An augmented fully-mixed finite element method for a coupled flow-transport problem

In this paper we analyze the coupling of the Stokes equations with a transport problem modelled by a scalar nonlinear convection–diffusion problem, where the viscosity of the fluid and the diffusion coefficient depend on the solution to the transport problem and its gradient, respectively. An augmented mixed variational formulation for both the fluid flow and the transport model is proposed. As a consequence, no discrete inf-sup conditions are required for the stability of the associated Galerkin scheme, and therefore arbitrary finite element subspaces can be used, which constitutes one of the main advantages of the present approach. In particular, the resulting fully-mixed finite element method can employ Raviart–Thomas spaces of order k for the Cauchy stress, continuous piecewise polynomials of degree $$k + 1$$ for the velocity and for the scalar field, and discontinuous piecewise polynomial approximations for the gradient of the concentration. In turn, the Lax–Milgram lemma, monotone operators theory, and the classical Schauder and Brouwer fixed point theorems are utilized to establish existence of solution of the continuous and discrete formulations. In addition, suitable estimates, arising from the combined use of a regularity assumption with the Sobolev embedding and Rellich–Kondrachov compactness theorems, are also required for the continuous analysis. Then, sufficiently small data allow us to prove uniqueness of solution and to derive optimal a priori error estimates. Finally, several numerical tests, illustrating the performance of our method and confirming the predicted rates of convergence, are reported.

Predictor–Corrector Nodal Integral Method for simulation of high Reynolds number fluid flow using larger time steps in Burgers ’ equation

Nodal Integral Methods (NIM), although frequently used for solving neutron transport equations, have not found wider acceptance for solving fluid flow problems. One of the pivotal reasons behind this is the lack of efficient non-linear solvers for the system of coupled algebraic equations obtained by these methods, thus making such methods prohibitively expensive for highly nonlinear (higher Reynolds number) fluid flow problems. Successive improvements in earlier attempts resulted in an improved version called Modified NIM (MNIM) scheme; following which, a further modified version appeared (called as M2NIM), that used the concept of the delayed coefficients. Upon comparing the solutions obtained using MNIM and M2NIM schemes, respectively, it was observed that although the modified scheme with delayed coefficients (M2NIM) has significantly faster convergence, it is less accurate. The convergence and accuracy are very likely to suffer to a significant extent in the high Reynolds number fluid flows, especially when a large time-step is to be considered. In order to resolve these difficulties, a new type of physics-based predictor–corrector algorithm is proposed in the present study. In the proposed computational algorithm, the linearized M2NIM scheme is used to predict the solution, whereas the MNIM scheme is utilized to improve the predictive guess. The novelty of such a hybrid numerical algorithm comes from the fact that it combines the advantage of the faster convergence of M2NIM with the accuracy of the MNIM scheme. The algorithm also benefits from the unique inclusion of Jacobian-free Newton-Krylov (JFNK) method, which helps in getting rid of the formation of large Jacobian matrices, thereby reducing unnecessary computational overhead. The proposed methodology has been applied to solve a non-linear convection–diffusion problem, represented by the Burgers’ equation. The computational results for bothone-dimensional and two-dimensional Burgers’ equation are presented to demonstrate the effectiveness of the developed novel algorithm, as well as, the advantage it offers over the existing numerical methods.

New corrective scheme for DF-SPH

A new corrective scheme for the derivative-free (DF) smoothed particle hydrodynamics (SPH) method is proposed to compute the first- and second-order derivatives. To demonstrate the ability of the new corrective scheme, the first- and second-order derivatives of the given function were computed when the particles were uniformly and irregularly distributed. The nonlinear convection–diffusion problems were simulated using both the DF-SPH method and the proposed method in one- and two-dimensional cases, and root mean square errors were compared. Results show that numerical solutions acquired by the proposed corrective scheme agree very well with analytical solutions and that the new scheme is more efficient, accurate, and stable than the original DF-SPH.

A review of hybrid integral transform solutions in fluid flow problems with heat or mass transfer and under Navier–Stokes equations formulation

The Generalized Integral Transform Technique (GITT) is reviewed as a hybrid numerical–analytical approach for fluid flow problems, with or without heat and mass transfer, here with emphasis on the literature related to flow problems formulated through the full Navier–Stokes equations. A brief overview of the integral transform methodology is first provided for a general nonlinear convection–diffusion problem. Then, different alternatives of eigenfunction expansion strategies are discussed in the integral transformation of problems for which the fluid flow model is either based on the primitive variables or the streamfunction-only formulations, as applied to both steady and transient states. Representative test cases are selected to illustrate the different eigenfunction expansion approaches, with convergence being analyzed for each situation. In addition, fully converged integral transform results are critically compared to previously reported simulations obtained from traditional purely discrete methods.

A Weak Galerkin Finite Element Method for Solving Nonlinear Convection–Diffusion Problems in One Dimension

We propose a weak Galerkin (WG) finite element method for solving one-dimensional nonlinear convection–diffusion problems. Based on a weak form, the semi-discrete WG finite element scheme is established and analyzed. We prove the stability of the semi-discrete solution and derive the optimal order error estimate in the discrete $$H^1$$ -norm and $$L^2$$ -norm, respectively. Numerical experiment is conducted to confirm the theoretical results.

A weak Galerkin finite element method for solving nonlinear convection-diffusion problems in two dimensions

We study weak Galerkin (WG) finite element method (FEM) for solving nonlinear convection-diffusion problems. A WG finite element scheme is presented based on a new variational form. We prove the energy conservation law and stability of the continuous time WG FEM. In particular, optimal order error estimates are established for the WG FEM approximation in both a discrete H1-norm and L2-norm. Numerical experiments are performed to confirm the theoretical results.

Stability of the ale space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains

The paper is concerned with the analysis of the space-time discontinuous Galerkin method (STDGM) applied to the numerical solution of nonstationary nonlinear convection-diffusion initial- boundary value problem in a time-dependent domain. The problem is reformulated using the arbitrary Lagrangian-Eulerian (ALE) method, which replaces the classical partial time derivative by the so-called ALE derivative and an additional convective term. The problem is discretized with the use of the ALE- space time discontinuous Galerkin method (ALE-STDGM). In the formulation of the numerical scheme we use the nonsymmetric, symmetric and incomplete versions of the space discretization of diffusion terms and interior and boundary penalty. The nonlinear convection terms are discretized with the aid of a numerical flux. The main attention is paid to the proof of the unconditional stability of the method. An important step is the generalization of a discrete characteristic function associated with the approximate solution and the derivation of its properties.

Accelerating the iterative solution of convection–diffusion problems using singular value decomposition

SummaryThe discretization of convection–diffusion equations by implicit or semi‐implicit methods leads to a sequence of linear systems usually solved by iterative linear solvers such as the generalized minimal residual method. Many techniques bearing the name of recycling Krylov space methods have been proposed to speed up the convergence rate after restarting, usually based on the selection and retention of some Arnoldi vectors. After providing a unified framework for the description of a broad class of recycling methods and preconditioners, we propose an alternative recycling strategy based on a singular value decomposition selection of previous solutions and exploit this information in classical and new augmentation and deflation methods. The numerical tests in scalar nonlinear convection–diffusion problems are promising for high‐order methods.

Optimal error estimate of the local discontinuous Galerkin methods based on the generalized alternating numerical fluxes for nonlinear convection–diffusion equations

In this paper, we present the optimal L2-norm error estimate of the local discontinuous Galerkin method based on the generalized alternating numerical flux for nonlinear convection–diffusion problem. We obtain the optimal (k + 1)-th error estimate for the piecewise tensor product polynomials space of degree at most k defined on any quasi-uniform Cartesian meshes. The highlight of the analysis is the use of some suitable designed global projections. Numerical experiments are given to verify the sharpness of the theoretical results.

Asymptotically Exact Posteriori Error Estimates for the Local Discontinuous Galerkin Method Applied to Nonlinear Convection–Diffusion Problems

In this paper, we present and analyze implicit a posteriori error estimates for the local discontinuous Galerkin (LDG) method applied to nonlinear convection–diffusion problems in one space dimension. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the $$L^2$$ -norm for the semi-discrete formulation. More precisely, we identify special numerical fluxes and a suitable projection of the initial condition for the LDG scheme to achieve $$p+1$$ order of convergence for the solution and its spatial derivative in the $$L^2$$ -norm, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order $$p+1$$ towards the derivative of a special projection of the exact solution. We use this result to prove that the LDG solution is superconvergent with order $$p+3/2$$ towards a special Gauss–Radau projection of the exact solution. Our superconvergence results allow us to show that the leading error term on each element is proportional to the $$(p+1)$$ -degree right Radau polynomial. We use these results to construct asymptotically exact a posteriori error estimator. Furthermore, we prove that the a posteriori LDG error estimate converges at a fixed time to the true spatial error in the $$L^2$$ -norm at $$\mathcal {O}(h^{p+3/2})$$ rate. Finally, we prove that the global effectivity index in the $$L^2$$ -norm converge to unity at $$\mathcal {O}(h^{1/2})$$ rate. Our proofs are valid for arbitrary regular meshes using $$P^p$$ polynomials with $$p\ge 1$$ . Finally, several numerical examples are given to validate the theoretical results.

Recent advances in computational-analytical integral transforms for convection-diffusion problems

An unifying overview of the Generalized Integral Transform Technique (GITT) as a computational-analytical approach for solving convection-diffusion problems is presented. This work is aimed at bringing together some of the most recent developments on both accuracy and convergence improvements on this well-established hybrid numerical-analytical methodology for partial differential equations. Special emphasis is given to novel algorithm implementations, all directly connected to enhancing the eigenfunction expansion basis, such as a single domain reformulation strategy for handling complex geometries, an integral balance scheme in dealing with multiscale problems, the adoption of convective eigenvalue problems in formulations with significant convection effects, and the direct integral transformation of nonlinear convection-diffusion problems based on nonlinear eigenvalue problems. Then, selected examples are presented that illustrate the improvement achieved in each class of extension, in terms of convergence acceleration and accuracy gain, which are related to conjugated heat transfer in complex or multiscale microchannel-substrate geometries, multidimensional Burgers equation model, and diffusive metal extraction through polymeric hollow fiber membranes. Numerical results are reported for each application and, where appropriate, critically compared against the traditional GITT scheme without convergence enhancement schemes and commercial or dedicated purely numerical approaches.

Superconvergence of the local discontinuous Galerkin method for nonlinear convection-diffusion problems

In this paper, we discuss the superconvergence of the local discontinuous Galerkin methods for nonlinear convection-diffusion equations. We prove that the numerical solution is (k+3/2)th-order superconvergent to a particular projection of the exact solution, when the upwind flux and the alternating fluxes are used. The proof is valid for arbitrary nonuniform regular meshes and for piecewise polynomials of degree k (kgeq1). The numerical experiments reveal that the property of superconvergence actually holds true for general fluxes.