Convection-dominated Convection-diffusion Equations Research Articles

An efficient variational multiscale element free Galerkin method based on adaptive TR-AB2 scheme for convection–diffusion equations

The variational multiscale element-free Galerkin (VMEFG) method can effectively improve stability when solving convection-dominated problems. However, for time-dependent problems, the VMEFG method has a large amount of calculation. In this paper, an efficient VMEFG method for solving unsteady convection-dominated convection–diffusion equations is proposed. The trapezoidal rule (TR) with adaptive time step is used for time integration, and the explicit second-order Adams–Bashforth (AB2) method is used to control the error. The main idea is to allow the time step to change along the direction of the solution, so that the amount of calculation under the specified accuracy requirements can be minimized. We further exemplified the effectiveness and accuracy of the given method in several examples. The numerical results show that the VMEFG method combined with TR-AB2 method (VMEFG-TR-AB2) can effectively increase computing efficiency while maintaining the calculation accuracy for the convection-dominated problem.

Using Deep Neural Networks for Detecting Spurious Oscillations in Discontinuous Galerkin Solutions of Convection-Dominated Convection–Diffusion Equations

Standard discontinuous Galerkin finite element solutions to convection-dominated convection–diffusion equations usually possess sharp layers but also exhibit large spurious oscillations. Slope limiters are known as a post-processing technique to reduce these unphysical values. This paper studies the application of deep neural networks for detecting mesh cells on which slope limiters should be applied. The networks are trained with data obtained from simulations of a standard benchmark problem with linear finite elements. It is investigated how they perform when applied to discrete solutions obtained with higher order finite elements and to solutions for a different benchmark problem.

Superconsistent collocation methods with applications to convection-dominated convection–diffusion equations

Superconsistency, as it applies to collocation schemes for linear partial differential equations, is given what we believe to be a new and precise definition. We introduce new superconsistent collocation schemes for linear partial differential equations and illustrate their use for the solution of two and three dimensional steady convection-dominated convection–diffusion equations, where the exact solutions exhibits exponential boundary layers near the outflow boundary of the convection velocity field. The numerical results reveal that our methods are more accurate than both a centred finite difference scheme and an existing upwinded scheme due to Fatone et al. (2005) over a range of convection velocities and grid sizes.

A high-order finite volume scheme for unsteady convection-dominated convection–diffusion equations

A finite volume scheme with high-order accuracy is proposed for solving the unsteady convection-dominated transport problems. In this scheme, some weighting parameters are introduced in discretizing both the convection term and time integral. These parameters are determined analytically by making the truncation error of the proposed scheme as small as possible. Since the discretization equations of the proposed scheme share the same band structure as that of the traditional finite volume method based on the central differencing scheme, the proposed scheme does not increase computing cost. Numerical results show that the proposed scheme not only can achieve sixth-order accuracy but also avoids any unphysical oscillations in the steep gradient region of the solutions of the linear and nonlinear convection-dominated convection–diffusion problems.

Anisotropic Meshes and Stabilization Parameter Design of Linear SUPG Method for 2D Convection-Dominated Convection–Diffusion Equations

We propose a numerical strategy to generate a sequence of anisotropic meshes and select appropriate stabilization parameters simultaneously for linear SUPG method solving two dimensional convection-dominated convection–diffusion equations. Since the discretization error in a suitable norm can be bounded by the sum of interpolation error and its variants in different norms, we replace them by some terms which contain the Hessian matrix of the true solution, convective field, and the geometric properties such as directed edges and the area of triangles. Based on this observation, the shape, size and equidistribution requirements are used to derive corresponding metric tensor and stabilization parameters. Numerical results are provided to validate the stability and efficiency of the proposed numerical strategy.

A new projection-based stabilized method for steady convection-dominated convection–diffusion equations

We propose a new projection-based stabilized method for the steady convection-dominated convection–diffusion equations. This method directly increases the L2-stability instead of the H1-stability which the variational multiscale (VMS) methods do. Numerical analysis and numerical experiments illustrate and confirm that our new method has effective numerical performances on convection-dominated convection–diffusion problems.

A Direct Solver with $O(N)$ Complexity for Variable Coefficient Elliptic PDEs Discretized via a High-Order Composite Spectral Collocation Method

A numerical method for solving elliptic PDEs with variable coefficients on two-dimensional domains is presented. The method is based on high-order composite spectral approximations and is designed for problems with smooth solutions. The resulting system of linear equations is solved using a direct (as opposed to iterative) solver that has optimal $O(N)$ complexity for all stages of the computation when applied to problems with nonoscillatory solutions such as the Laplace and the Stokes equations. Numerical examples demonstrate that the scheme is capable of computing solutions with a relative accuracy of $10^{-10}$ or better for challenging problems such as highly oscillatory Helmholtz problems and convection-dominated convection-diffusion equations. In terms of speed, it is demonstrated that a problem with a nonoscillatory solution that was discretized using $10^{8}$ nodes can be solved in 115 minutes on a personal workstation with two quad-core 3.3 GHz CPUs. Since the solver is direct, and the “solution ...

An assessment of discretizations for convection-dominated convection–diffusion equations

The performance of several numerical schemes for discretizing convection-dominated convection–diffusion equations will be investigated with respect to accuracy and efficiency. Accuracy is considered in measures which are of interest in applications. The study includes an exponentially fitted finite volume scheme, the Streamline-Upwind Petrov–Galerkin (SUPG) finite element method, a spurious oscillations at layers diminishing (SOLD) finite element method, a finite element method with continuous interior penalty (CIP) stabilization, a discontinuous Galerkin (DG) finite element method, and a total variation diminishing finite element method (FEMTVD). A detailed assessment of the schemes based on the Hemker example will be presented.

On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part II – Analysis for [formula omitted] and [formula omitted] finite elements

An unwelcome feature of the popular streamline upwind/Petrov–Galerkin (SUPG) stabilization of convection-dominated convection–diffusion equations is the presence of spurious oscillations at layers. A review and a comparison of the most methods which have been proposed to remove or, at least, to diminish these oscillations without leading to excessive smearing of the layers are given in Part I, [V. John, P. Knobloch, On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part I – A review, Comput. Methods Appl. Mech. Engrg. 196 (2007) 2197–2215]. In the present paper, the most promising of these SOLD methods are investigated in more detail for P 1 and Q 1 finite elements. In particular, the dependence of the results on the mesh, the data of the problems and parameters of the methods are studied analytically and numerically. Furthermore, the numerical solution of the nonlinear discrete problems is discussed and the capability of adaptively refined grids for reducing spurious oscillations is examined. Our conclusion is that, also for simple problems, any of the SOLD methods generally provides solutions with non-negligible spurious oscillations.

On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part I – A review

An unwelcome feature of the popular streamline upwind/Petrov–Galerkin (SUPG) stabilization of convection-dominated convection–diffusion equations is the presence of spurious oscillations at layers. Since the mid of the 1980s, a number of methods have been proposed to remove or, at least, to diminish these oscillations without leading to excessive smearing of the layers. The paper gives a review and state of the art of these methods, discusses their derivation, proposes some alternative choices of parameters in the methods and categorizes them. Some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods.

A two-level variational multiscale method for convection-dominated convection–diffusion equations

This paper studies the error in, the efficient implementation of and time stepping methods for a variational multiscale method (VMS) for solving convection-dominated problems. The VMS studied uses a fine mesh C0 finite element space Xh to approximate the concentration and a coarse mesh discontinuous vector finite element space LH for the large scales of the flux in the two scale discretization. Our tests show that these choices lead to an efficient VMS whose complexity is further reduced if a (locally) L2-orthogonal basis for LH is used. A fully implicit and a semi-implicit treatment of the terms which link effects across scales are tested and compared. The semi-implicit VMS was much more efficient. The observed global accuracy of the most straightforward VMS implementation was much better than the artificial diffusion stabilization and comparable to a streamline-diffusion finite element method in our tests.

Local Error Estimates for a Finite Element Method for Hyperbolic and Convection-Diffusion Equations

Local error estimates of near optimal order are derived for a finite element method for hyperbolic and convection dominated convection-diffusion equations in a domain $\Omega \subset R^2 $. The method generates, in an explicit fashion, a continuous piecewise polynomial approximation of degree $n \geqslant 2$ over a triangulation of $\Omega $ The scheme is shown to propagate disturbances a distance $O(\sqrt h \log \frac{1} {h})$ in the crosswind direction, where h is the meshsize. The analysis uses test functions which depend only on the crosswind variable. It is also shown to be applicable, in a parallel fashion, to the discontinuous Galerkin method, thus underscoring the close interrelationship of the two methods.

Particle-method solution of two-dimensional convection-diffusion equations

We present a new method for solving two-dimensional convection-dominated convection-diffusion equations containing spatially and temporally localized source terms. The method uses grid-free transport: particles which carry point values of the concentration gradient move by convection and undergo random-walk to simulate diffusion. No numerical diffusion is introduced. Equations for the evolution of the gradient values are solved. The concentration is recovered from the particle data by solving Poisson equations. The method is applied to problems in rotational and elongational flow fields. Numerical results demonstrating convergence of the method are presented.