Convection-dominated Problems Research Articles

An adaptive variational multiscale element free Galerkin method based on the residual-based a posteriori error estimators for convection–diffusion–reaction problems

As we know, the variational multiscale element free Galerkin (VMEFG) method may still suffer from non-physical oscillations near the boundary or interior layers when solving the convection–diffusion–reaction problems with strong convection-dominated. To overcome this shortcoming, we consider incorporating an adaptive algorithm based on the residual error estimations into the VMEFG, which formed the adaptive VMEFG (AVMEFG) method. With respect to the adaptive technique adopted, two residual-based a posteriori error estimators in the H1-semi norm and energy norm are respectively used to locate and remark the high-gradient numerical solution regions. Several stationary convection–diffusion–reaction problems are solved to verify the effectiveness of the proposed method. Among them, the first example makes a comparison between the proposed method and the adaptive element free Galerkin method, while the other remaining examples compare two different residual-based a posteriori error estimators for the proposed method. Numerical examples illustrate that the proposed method is effective and efficient in solving the convection-dominated problem involving various layers. Moreover, the AVMEFG methods with the two residual-based a posteriori error estimators are generally equivalent except in the case that the solution has an exponential boundary layer, where the energy norm error estimator can achieve significantly better results than the other one.

A semi-Lagrangian meshfree Galerkin method for convection-dominated partial differential equations

A new numerical method, which is based on the coupling between semi-Lagrangian (SL) method and element free Galerkin (EFG) method, is developed for convection–diffusion partial differential equations with dominated convection terms. It is the first time that the SL method combined with the EFG method to solve the transient convection–diffusion equations. This combination has been more effective than expected. Compared with the existing SL methods, the proposed method shows several new advantages. Firstly, it takes full advantages of meshfree methods, therefore, the implementation difficulties of the SL method in backward tracing and departure point interpolation are greatly reduced. As interpolating the function values to a departure point, complex searching algorithm is not needed to determine which grid cell contains the departure point, and higher order interpolation is not essential to ensure the global accuracy. Meanwhile, in addition to having good stability and allowing the use of large time-step sizes, the new method realizes the continuous transition from convection–diffusion equation to pure convection equation and provides a uniform numerical format for these two kinds of equations. A problem with available analytical solution is solved to analyze the convergence behavior of the proposed method. It is demonstrated that the method can achieve the optimal convergence rate. Subsequently, the method is applied to solve the 1D and 2D Burgers’ equations. The results show that the proposed method can indeed obtain surprisingly good results for the Burgers’ equations in the very challenging convection-dominated and pure convection cases. These results are better than almost all the ones we have seen in literature. The good performance of this method in solving the challenging Burgers’ equations makes us believe that it will be a very promising method for solving convection-dominated problems.

A half boundary method for two dimensional unsteady convection–diffusion equations

Convection diffusion equation is widely applied in many fields of science and technology. Many practical engineering problems can be expressed by this equation in unsteady state. However, it is usually difficult and time-consuming to find its solution. In this paper, a boundary type method named half boundary method (HBM) is proposed for two dimensional unsteady convection–diffusion equations. The main idea of HBM is introducing new variables to reduce the order of the equation and build relations of variables between nodes inside the area and nodes at half of the boundaries, namely the unknown variables. Since only variables at half of the boundaries are chosen as unknown variables, using HBM can realize dimensionality reduction and the maximum matrix order is less than that in finite volume method when considering large division number, which makes HBM fast and efficient. After the unknown variables are obtained by the boundary conditions, all variables at nodes can be obtained simultaneously according to the relations mentioned above. Numerical studies are carried out for convection dominated problems, problems with variable or discontinuous coefficient and problems with mixed boundary conditions, which show the validity of HBM for two-dimensional unsteady convection-diffusion problems and high accuracy for convection-domination problems.

A stabilized finite volume element method for solving Poisson–Nernst–Planck equations

One difficulty in solving the Poisson-Nernst-Planck (PNP) equations used for studying the ion transport in channel proteins is the possible convection-dominant problem in the Nernst-Planck equations. In this paper, to overcome this issue, considering the general mixed boundary conditions of concentration functions on the interface, a novel stabilized finite volume element method based on the standard weak formulation to solve the steady-state PNP equations is proposed and analyzed. Numerical tests on four ion-channel proteins served as benchmark with varying boundary conditions in a certain range show that the new stabilized technique not only improves the robustness of the new PNP solver, but also makes the computed (especially the maximal) concentration values much more reasonable.

A Galerkin-Characteristic Finite Element Method for Three-Dimensional Convection-Dominated Problems

A Galerkin-Characteristic Finite Element Method for Three-Dimensional Convection-Dominated Problems

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.

Numerical Simulation of the Convection–Diffusion PDEs on a Sphere with RBF-FD and RBF-QR Methods

The aim of this paper is to investigate the application of radial basis function-generated finite difference (RBF-FD) methods for convection–diffusion partial differential equations (PDEs) on a sphere. In the application of RBF-FD method, choosing a reasonable value of shape parameter is important to the computation of PDEs. The work is devoted to the numerical study of the range of near optimal shape parameters for the convection–diffusion equations. Because the RBF-FD Direct method often leads to ill-conditioned problems for small shape parameters, the RBF-QR method is applied locally to overcome the ill-conditioning in the context of RBF-FD mode. Additionally, for convection-dominated problems, it can be found that the results of using central-type stencil present spurious oscillations. Therefore, we propose an upwind RBF-FD (URBF-FD) scheme to overcome the problems, which is well adapted to the problems on the sphere and easy to be implemented. Further numerical results show that the proposed URBF-FD method is stable and effective for convection-dominated PDEs on the sphere.

Gradient recovery-based adaptive stabilized mixed FEM for the convection–diffusion–reaction equation on surfaces

In this article, we consider a gradient recovery-based adaptive stabilized mixed finite element method (FEM) for solving the convection–diffusion–reaction equation on surfaces. By defining a new total flux on the surface, we firstly transform the original equation into an equivalent mixed first-order formulation. Then, by using both the convection–convection term and the div–div term to stabilize the weak formulation, we prove the discrete inf–sup condition and the error estimate for the mixed FEM. In this new method, the equal order Lagrangian elements are allowed for the primal variable and the total flux, and a high accurate numerical solution for the flux in the H1 space is obtained. The impact of the curvature on the algorithm is also investigated. Additionally, to overcome the low resolution and the non-physical oscillation in simulating the convection-dominated problem, we study a gradient recovery-based adaptive stabilized mixed FEM. Finally, we show some numerical experiments to verify the theoretical prediction and efficiency of the studied method.

Importance of parameter optimization in a nonlinear stabilized method adding a crosswind diffusion

Numerical solutions of convection-dominated problems are known to exhibit spurious oscillations whose suppression requires the use of numerical stabilization. Stabilized methods which involve heuristic parameters are often applied. The parameters influence the quality of the solution but their optimal values are unknown.In this paper, we consider a stabilization method which adds numerical diffusion adaptively, based on minimizing a functional. The novelty of our approach consists in combining an error indicator with reduced residuals with a nonlinear SOLD method adding artificial diffusion in the crosswind direction. We demonstrate that this approach can lead to more physically meaningful solutions than techniques considered before.

The Galerkin least squares MLPG method for convection-dominated problems

The meshless method has been widely applied in the computational mechanics, materials science, computational heat transfer and fluid flow. However, the development of a high efficient method for convection term in the computational fluid dynamics is difficult. In this paper, a truly meshless method of the meshless local Petrov-Galerkin (MLPG) method was applied to solve convection-diffusion problem. Meanwhile, the Galerkin least squares (GLS) approximation method was proposed to overcome the influence of convection term. The accuracy and efficiency of the present method were validated by some cases with benchmark solutions. The computational results showed that the GLS method could be used in dealing with convection-diffusion problems with high computational precision.

The Galerkin least squares MLPG method for convection-dominated problems

The meshless method has been widely applied in the computational mechanics, materials science, computational heat transfer and fluid flow. However, the development of a high efficient method for convection term in the computational fluid dynamics is difficult. In this paper, a truly meshless method of the meshless local Petrov-Galerkin (MLPG) method was applied to solve convection-diffusion problem. Meanwhile, the Galerkin least squares (GLS) approximation method was proposed to overcome the influence of convection term. The accuracy and efficiency of the present method were validated by some cases with benchmark solutions. The computational results showed that the GLS method could be used in dealing with convection-diffusion problems with high computational precision.

Spatio-stochastic adaptive discontinuous Galerkin methods

We introduce a goal-oriented, adaptive framework for the uncertainty quantification of systems modeled by stochastically parametrized nonlinear hyperbolic and convection-dominated partial differential equations. Of particular interest are conservation laws in aerodynamics that may have a small number of stochastic parameters but exhibit strong nonlinearity and a wide range of scales. Our framework exploits localized structure in the spatio-parameter space to enable rapid, reliable uncertainty quantification for output quantities of interest. Our formulation comprises the following technical components: (i) a discontinuous Galerkin finite element method, which provides stability for convection-dominated problems; (ii) element-wise polynomial chaos expansions, which capture the parametric dependence of the solution in a way amenable to adaptation; (iii) the dual-weighted residual method, which provides global and element-wise error estimates for quantities of interest; and (iv) a projection-based anisotropic error indicator along with the associated adaptation mechanics that can detect and refine strongly directional features in the physical and/or parameter spaces simultaneously in an efficient manner. Both the spatial and stochastic discretization errors are controlled through the adaptive refinement of the spatial mesh or polynomial chaos expansion degree based on these anisotropic error indicators. We analyze stability, approximation properties, and a priori error bounds of the spatio-stochastic adaptive method. We finally demonstrate the effectiveness of our formulation for engineering-relevant transonic turbulent aerodynamics problems with uncertainties in flow conditions and turbulence parameters.

A low-dissipation shock-capturing framework with flexible nonlinear dissipation control

In this work, a framework to construct arbitrarily high-order low-dissipation shock-capturing schemes with flexible and controllable nonlinear dissipation for convection-dominated problems is proposed. While a set of candidate stencils of incremental width is constructed, each one is indicated as smooth or nonsmooth by the ENO-like stencil selection procedure proposed in the targeted essentially non-oscillatory (TENO) scheme (Fu et al. 2016 [9]). Rather than being discarded directly as with TENO schemes, the nonsmooth candidates are filtered by an extra nonlinear limiter, such as a monotonicity-preserving (MP) limiter or a total variation diminishing (TVD) limiter. Consequently, high-order reconstruction is achieved by assembling candidate fluxes with optimal linear weights since they are either smooth reconstructions or filtered ones which feature good non-oscillation property. A weight renormalization procedure as with the standard TENO paradigm is not necessary. This new framework concatenates the concepts of TENO, WENO and other nonlinear limiters for shock-capturing, and provides a new insight to designing low-dissipation nonlinear schemes. Through the adaptation of nonlinear limiters, nonlinear dissipation in the newly proposed framework can be controlled separately without affecting the performance in smooth regions. Based on the proposed framework, a family of new six- and eight-point nonlinear schemes with controllable dissipation is proposed. A set of critical benchmark cases involving strong discontinuities and broadband fluctuations is simulated. Numerical results reveal that the proposed new schemes capture discontinuities sharply and resolve the high-wavenumber fluctuations with low dissipation, while maintaining the desired accuracy order in smooth regions.

A mass-conservative higher-order ADI method for solving unsteady convection\u2013diffusion equations

In the paper, a high-order alternating direction implicit (ADI) algorithm is presented to solve problems of unsteady convection and diffusion. The method is fourth- and second-order accurate in space and time, respectively. The resulting matrix at each ADI computation can be obtained by repeatedly solving a penta-diagonal system which produces a computationally cost-effective solver. We prove that the proposed scheme is mass-conserved and unconditionally stable by means of discrete Fourier analysis. Numerical experiments are performed to validate the mass conservation and illustrate that the proposed scheme is accurate and reliable for convection-dominated problems.

Collocated Finite-Volume Method for the Incompressible Navier-Stokes Problem

Abstract The article introduces a collocated finite-volume method for the incompressible Navier-Stokes problem. The method applies to general polyhedral grids and demonstrates higher than the first order of convergence. The velocity components and the pressure are approximated by piecewise-linear continuous and piecewise-constant fields, respectively. The method does not require artificial boundary conditions for pressure but requires stabilization term to suppress the error introduced by piecewise-constant pressure for convection-dominated problems. Both the momentum and continuity equations are approximated in a flux-conservative fashion, i.e. the conservation for both quantities is discretely exact. The attractive side of the method is a simple flux-based finite-volume construction of the scheme. Applicability of the method is demonstrated on several numerical tests using general polyhedral grids.

A Hybrid Time Integration Scheme for the Discontinuous Galerkin Discretizations of Convection-Dominated Problems

Discontinuous Galerkin (DG) method is a popular high-order accurate method for solving unsteady convection-dominated problems. After spatially discretizing the problem with the DG method, a time integration scheme is necessary for evolving the result. Owing to the stability-based restriction, the time step for an explicit scheme is limited by the smallest element size within the mesh, making the calculation inefficient. In this paper, a hybrid scheme comprising a three-stage, third-order accurate, and strong stability preserving Runge–Kutta (SSP-RK3) scheme and the three-stage, third-order accurate, L-stable, and diagonally implicit Runge–Kutta (LDIRK3) scheme is proposed. By dealing with the coarse and the refined elements with the explicit and implicit schemes, respectively, the time step for the hybrid scheme is free from the limitation of the smallest element size, making the simulation much more efficient. Numerical tests and comparison studies were made to show the performance of the hybrid scheme.

Upwind skewed radial basis functions (USRBF) for solution of highly convective problems over meshfree nodes

An upwind skewed radial basis function (USRBF)-based solution scheme is presented for stabilized solutions of convection-dominated problems over meshfree nodes. The conventional, radially symmetric radial basis functions (RBFs) are multiplied with an upwinding factor which skews the RBFs toward the upwind direction. The upwinding factor is a function of flow direction, intensity of convection, size of local support domain, and nodal distribution. The use of USRBFs modifies the weight values such that the necessary artificial diffusion is added only along the flow direction, whereas the crosswind diffusion is avoided. Subsequently, these skewed radial basis functions are employed in finite difference mode (RBF-FD) for derivative approximation. The performance and accuracy of the proposed scheme is studied by solving convection–diffusion problems over uniform and random distribution of meshfree nodes with various convection intensities. The upwinding effectively suppresses non-physical perturbation in numerical solution of convection-dominated problems. The results show that significant improvement in accuracy can be achieved by using the proposed USRBF-based solution scheme, particularly at higher convection intensities.

A modified monotonicity-preserving high-order scheme with application to computation of multi-phase flows

The monotonicity preserving (MP) polynomial-based reconstruction scheme with Runge-Kutta time stepping by Suresh and Huynh (1997) has been applied extensively in computational fluid dynamics for solving hyperbolic partial differential equations and other convection-dominated problems. However, despite of its algorithmic simplicity in achieving a desired high-order of accuracy, the MP scheme is shown to lose the accuracy and robustness for a long-term computation of flows containing complex smooth structures interspersed with discontinuities. In this paper, the original MP scheme is first reviewed with a focus on the potential causes that lead to its failure. We then propose an improved MP scheme that can overcome the weakness in the MP scheme. The proposed MP scheme is constructed so that it can bypass smooth extrema without loss of accuracy while preserving monotonicity and properly reducing the order of accuracy near discontinuities. Various challenging scalar advection test cases are considered to demonstrate the performance of the presented method. Numerical results reveal that it achieves the highly accurate solutions in smooth parts of the flow and yields a significant improvement in accuracy and shape-preserving property. Moreover, the successful application to several classes of two- and three-dimensional unsteady multi-phase flows highlights the potential of the present method for resolving both sharp and complex interface deformations.

Appropriate stabilized Galerkin approaches for solving two-dimensional coupled Burgers’ equations at high Reynolds numbers

This paper aims to seek proper stabilized Galerkin methods for solving the two-dimensional coupled Burgers’ equations at high Reynolds numbers. The stabilization techniques employed here include the streamline upwind/Petrov–Galerkin (SUPG) method, the spurious oscillations at layers diminishing (SOLD) method and the characteristic Galerkin (CG) method. The first two methods are combined with the Crank–Nicolson scheme for time discretization and the last one is applied in its semi-implicit version. Different from most of the studies on the equations which are usually devoted to improving the accuracy of computed solution in the case of low Reynolds numbers, this paper mainly focuses on keeping the stability of the solution at high Reynolds numbers, which is significant in practical applications and also challenging in numerical computation. We study two problems, equipped with mixed boundary conditions and only Dirichlet boundary conditions, respectively. Numerical experiments reveal that the SUPG method is optimal for the former problem, and the SOLD method is more appropriate for the latter one. In addition, the performances of these methods demonstrate the difference between the two problems, which is seldom mentioned previously and might be helpful to other conventional methods intending to solve the problems at high Reynolds numbers. And last, since SOLD methods have rarely been utilized to solve nonlinear unsteady problems before, this study also indicates the potential of this class of methods to solve nonlinear unsteady convection-dominated problems.

A flux-corrected RBF-FD method for convection dominated problems in domains and on manifolds

Abstract In this work, we present a Flux-Corrected Transport (FCT) algorithm for enforcing discrete maximum principles in Radial Basis Function (RBF) generalized Finite Difference (FD) methods for convection-dominated problems. The algorithm is constructed to guarantee mass conservation and to preserve positivity of the solution for irregular data nodes. The method can be applied both for problems defined in a domain or if equipped with level set techniques, on a stationary manifold. We demonstrate the numerical behavior of the method by performing numerical tests for the solid-body rotation benchmark in a unit square and for a transport problem along a curve implicitly prescribed by a level set function. Extension of the proposed method to higher dimensions is straightforward and easily realizable.