Convection-dominated Transport Problems Research Articles

Puiseux asymptotic expansions for convection-dominated transport problems in thin graph-like networks: Strong boundary interactions

This article completes the study of the influence of the intensity parameter α in the boundary condition ε ∂ ν ε u ε − u ε V ε → · ν ε = ε α φ ε given on the boundary of a thin three-dimensional graph-like network consisting of thin cylinders that are interconnected by small domains (nodes) with diameters of order O ( ε ). Inside of the thin network a time-dependent convection-diffusion equation with high Péclet number of order O ( ε − 1 ) is considered. The novelty of this article is the case of α < 1, which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity φ ε (the cases α = 1 and α > 1 were previously studied by the same authors). A complete Puiseux asymptotic expansion is constructed for the solution u ε as ε → 0, i.e., when the diffusion coefficients are eliminated and the thin network shrinks into a graph. Furthermore, the corresponding uniform pointwise and energy estimates are proved, which provide an approximation of the solution with a given accuracy in terms of the parameter ε.

A bounded scheme based on Bézier curves for convection-dominated transport problems

This work presents a family of bounded schemes constructed based on the concept of Bézier curves that satisfies the TVD, CBC, and BAIR stability criteria for the numerical solution of convection-dominated transport problems in computational fluid dynamics problems. The convergence of the schemes to solve a series of 1D and 2D problems is analyzed. The numerical results of the proposed scheme are compared to other established methods from the literature. The numerical results show good agreement with exact solutions and numerical data from benchmarks. The findings of this work suggest the proposed family of methods as a promising alternative for solving convection-dominated problems.

Asymptotic approximations for semilinear parabolic convection-dominated transport problems in thin graph-like networks

We consider time-dependent convection-diffusion problems with high Péclet number of order O(ε−1) in thin three-dimensional graph-like networks consisting of cylinders that are interconnected by small domains (nodes) with diameters of order O(ε). On the lateral surfaces of the thin cylinders and the boundaries of the nodes we account for solution-dependent inhomogeneous Robin boundary conditions which render the associated initial-boundary problem to be nonlinear. The strength of the inhomogeneity is controlled by an intensity factor of order O(εα), α∈R. The asymptotic behavior of the solution is studied as ε→0, i.e., when the diffusion coefficients are eliminated and the thin three-dimensional network is shrunk into a graph. Three qualitatively different cases are discovered in the asymptotic behavior of the solution depending on the value of the intensity parameter α: α=1, α>1, and α<1. We construct the asymptotic approximation of the solution, which provides us with the hyperbolic limit model for ε→0 for the first two cases, and prove the corresponding uniform pointwise estimates and energy estimates. As the main result, we derive uniform pointwise estimates for the difference between the solutions of the convection-diffusion problem and the zero-order approximation that includes the solution of the corresponding hyperbolic limit problem.

Reprint of: Robust and globally efficient reduction of parametric, highly nonlinear computational models and real time online performance

This paper considers the problem of parametric, nonlinear, projection-based model order reduction (PMOR) and makes three contributions to advance its state of the art. First, it presents a computational strategy for reducing the total offline cost of nonlinear PMOR by enabling hyperreduction to be performed adaptively within the typical greedy sampling procedure, rather than uniformly at each of its iteration. Then, it presents a more accurate and computationally efficient approach for training hyperreduction based on residual Jacobians rather than residuals. Finally, it demonstrates for a large-scale, industrial-grade, parametric, nonlinear application, that the two aforementioned contributions make both offline and online stages of nonlinear PMOR practical in today’s engineering environments. The paper also shows that despite the so-called Kolmogorov n-width barrier for the model reduction of convection-dominated transport problems, the current state of the art of nonlinear, Petrov–Galerkin PMOR equipped with the advances proposed in this paper is capable of accelerating by orders of magnitude the accurate solution of shape-parametric, high Reynolds number, viscous flow problems represented by the Reynolds-averaged Navier–Stokes equations augmented with turbulence modeling. All three contributions are presented and discussed in the contexts of steady-state CFD problems; however, they are equally applicable to unsteady CFD problems as well as problems in solid mechanics and structural dynamics.

Robust and globally efficient reduction of parametric, highly nonlinear computational models and real time online performance

This paper considers the problem of parametric, nonlinear, projection-based model order reduction (PMOR) and makes three contributions to advance its state of the art. First, it presents a computational strategy for reducing the total offline cost of nonlinear PMOR by enabling hyperreduction to be performed adaptively within the typical greedy sampling procedure, rather than uniformly at each of its iteration. Then, it presents a more accurate and computationally efficient approach for training hyperreduction based on residual Jacobians rather than residuals. Finally, it demonstrates for a large-scale, industrial-grade, parametric, nonlinear application, that the two aforementioned contributions make both offline and online stages of nonlinear PMOR practical in today’s engineering environments. The paper also shows that despite the so-called Kolmogorov n-width barrier for the model reduction of convection-dominated transport problems, the current state of the art of nonlinear, Petrov–Galerkin PMOR equipped with the advances proposed in this paper is capable of accelerating by orders of magnitude the accurate solution of shape-parametric, high Reynolds number, viscous flow problems represented by the Reynolds-averaged Navier–Stokes equations augmented with turbulence modeling. All three contributions are presented and discussed in the contexts of steady-state CFD problems; however, they are equally applicable to unsteady CFD problems as well as problems in solid mechanics and structural dynamics.

Quadratic approximation manifold for mitigating the Kolmogorov barrier in nonlinear projection-based model order reduction

A quadratic approximation manifold is presented for performing nonlinear, projection-based, model order reduction (PMOR). It constitutes a departure from the traditional affine subspace approximation that is aimed at mitigating the Kolmogorov barrier for nonlinear PMOR, particularly for convection-dominated transport problems. It builds on the data-driven approach underlying the traditional construction of projection-based reduced-order models (PROMs); is application-independent; is linearization-free; and therefore is robust for highly nonlinear problems. Most importantly, this approximation leads to quadratic PROMs that deliver the same accuracy as their traditional counterparts using however a much smaller dimension – typically, n2∼n1, where n2 and n1 denote the dimensions of the quadratic and traditional PROMs, respectively. The computational advantages of the proposed high-order approach to nonlinear PMOR over the traditional approach are highlighted for the detached-eddy simulation-based prediction of the Ahmed body turbulent wake flow, which is a popular CFD benchmark problem in the automotive industry. For a fixed accuracy level, these advantages include: a reduction of the total offline computational cost by a factor greater than five; a reduction of its online wall clock time by a factor greater than 32; and a reduction of the wall clock time of the underlying high-dimensional model by a factor greater than two orders of magnitude.

An assessment of solvers for algebraically stabilized discretizations of convection–diffusion–reaction equations

Abstract We consider flux-corrected finite element discretizations of 3D convection-dominated transport problems and assess the computational efficiency of algorithms based on such approximations. The methods under investigation include flux-corrected transport schemes and monolithic limiters. We discretize in space using a continuous Galerkin method and ℙ1 or ℚ1 finite elements. Time integration is performed using the Crank–Nicolson method or an explicit strong stability preserving Runge–Kutta method. Nonlinear systems are solved using a fixed-point iteration method, which requires solution of large linear systems at each iteration or time step. The great variety of options in the choice of discretization methods and solver components calls for a dedicated comparative study of existing approaches. To perform such a study, we define new 3D test problems for time dependent and stationary convection–diffusion–reaction equations. The results of our numerical experiments illustrate how the limiting technique, time discretization and solver impact on the overall performance.

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.

Linearity-preserving monotone local projection stabilization schemes for continuous finite elements

This paper presents some new tools for enforcing discrete maximum principles and/or positivity preservation in continuous piecewise-linear finite element approximations to convection-dominated transport problems. Using a linear first-order advection equation as a model problem, we construct element-level bilinear forms associated with first-order artificial diffusion operators and their two-scale counterparts. The underlying design philosophy is similar to that behind local projection stabilization (LPS) techniques and variational multiscale (VMS) methods. The difference lies in the structure of the local stabilization operator and in the way in which the resolved scales are detected. The proposed stabilization term penalizes the difference between the nodal values and cell averages of the finite element solution in a manner which guarantees monotonicity and linearity preservation. The value of the stabilization parameter is determined using a multidimensional limiter function designed to prevent unresolvable fine scale effects from creating undershoots or overshoots. The result is a nonlinear high-resolution scheme capable of resolving moving fronts and internal/boundary layers as sharp localized nonoscillatory features. The use of variational gradient recovery makes it possible to add high-order background dissipation leading to improved approximation properties in smooth regions. The numerical behavior of the constrained schemes is illustrated by a grid convergence study for stationary and time-dependent test problems in two space dimensions.

Multiscale Simulations for Coupled Flow and Transport Using the Generalized Multiscale Finite Element Method

In this paper, we develop a mass conservative multiscale method for coupled flow and transport in heterogeneous porous media. We consider a coupled system consisting of a convection-dominated transport equation and a flow equation. We construct a coarse grid solver based on the Generalized Multiscale Finite Element Method (GMsFEM) for a coupled system. In particular, multiscale basis functions are constructed based on some snapshot spaces for the pressure and the concentration equations and some local spectral decompositions in the snapshot spaces. The resulting approach uses a few multiscale basis functions in each coarse block (for both the pressure and the concentration) to solve the coupled system. We use the mixed framework, which allows mass conservation. Our main contributions are: (1) the development of a mass conservative GMsFEM for the coupled flow and transport; (2) the development of a robust multiscale method for convection-dominated transport problems by choosing appropriate test and trial spaces within Petrov-Galerkin mixed formulation. We present numerical results and consider several heterogeneous permeability fields. Our numerical results show that with only a few basis functions per coarse block, we can achieve a good approximation.

Application of quadratures and operator splitting to partial differential equations

In this work we study the stability of a method for the numerical solution of initial value problems, that combines finite differences with Simpson's rule. This method is applied to a one spatial dimension, convection-dominated transport problem. To solve the same problem in two spatial dimensions, the proposed method was used in combination with Strang's operator decomposition method.

Quasi-Linear Convection-Dominated Transport Problem Based on Characteristics-Mixed Finite Element Method

We propose the characteristics-mixed method for approximating the solution to a convection-dominated transport problem. The new method is a combination of characteristic approximation to handle the convection part in time and a mixed finite element spatial approximation to deal with the diffusion part. Boundary conditions are incorporated in a natural fashion. The scheme is locally conservative since fluid is transported along the approximate characteristics on the discrete level and the test function can be piecewise constant. Analysis shows that the scheme has much smaller time-truncation errors than those of standard methods.

FINITE ELEMENT SOLUTION OF MULTI-SCALE TRANSPORT PROBLEMS USING THE LEAST SQUARES-BASED BUBBLE FUNCTION ENRICHMENT

This paper presents a technique for deriving least-squares-based polynomial bubble functions to enrich the standard linear finite elements, employed in the formulation of Galerkin weighted-residual statements. The element-level linear shape functions are enhanced using supplementary polynomial bubble functions with undetermined coefficients. The enhanced shape functions are inserted into the model equation and the residual functional is constructed and minimized by using the method of the least squares, resulting in an algebraic system of equations which can be solved to determine the unknown polynomial coefficients in terms of element-level nodal values. The stiffness matrices are subsequently formed with the standard finite elements assembly procedures followed by using these enriched elements which require no additional nodes to be introduced and no extra degree of freedom incurred. Furthermore, the proposed technique is tested on a number of benchmark linear transport equations where the quadratic and cubic bubble functions are derived and the numerical results are compared against the exact and standard linear element solutions. It is demonstrated that low order bubble enriched elements provide more accurate approximations for the exact analytical solutions than the standard linear elements at no extra computational cost in spite of using relatively crude meshes. On the other hand, it is observed that a satisfactory solution of the strongly convection-dominated transport problems may require element enrichment by using significantly higher order polynomial bubble functions in addition to the use of extremely fine computational meshes.

A bounded upwinding scheme for computing convection-dominated transport problems

A practical high resolution upwind differencing scheme for the numerical solution of convection-dominated transport problems is presented. The scheme is based on TVD and CBC stability criteria and is implemented in the context of the finite difference methodology. The performance of the scheme is investigated by solving the 1D/2D scalar advection equations, 1D inviscid Burgers’ equation, 1D scalar convection–diffusion equation, 1D/2D compressible Euler’s equations, and 2D incompressible Navier–Stokes equations. The numerical results displayed good agreement with other existing numerical and experimental data.

Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes

This paper is concerned with the development of general-purpose algebraic flux correction schemes for continuous (linear and multilinear) finite elements. In order to enforce the discrete maximum principle (DMP), we modify the standard Galerkin discretization of a scalar transport equation by adding diffusive and antidiffusive fluxes. The result is a nonlinear algebraic system satisfying the DMP constraint. An estimate based on variational gradient recovery leads to a linearity-preserving limiter for the difference between the function values at two neighboring nodes. A fully multidimensional version of this scheme is obtained by taking the sum of local bounds and constraining the total flux. This new approach to algebraic flux correction provides a unified treatment of stationary and time-dependent problems. Moreover, the same algorithm is used to limit convective fluxes, anisotropic diffusion operators, and the antidiffusive part of the consistent mass matrix.The nonlinear algebraic system associated with the constrained Galerkin scheme is solved using fixed-point defect correction or a nonlinear SSOR method. A dramatic improvement of nonlinear convergence rates is achieved with the technique known as Anderson acceleration (or Anderson mixing). It blends a number of last iterates in a GMRES fashion, which results in a Broyden-like quasi-Newton update. The numerical behavior of the proposed algorithms is illustrated by a grid convergence study for convection-dominated transport problems and anisotropic diffusion equations in 2D.

The use of quadratures for solving convective and highly stiff transport problems

In this paper we introduce a method for the numerical solutions of initial value problems, that combines finite differences with Simpson’s rule. The effectiveness of the method is proved by solving, in one spatial dimension, a stiff and convection-dominated transport problem. To solve the same problem in two spatial dimensions, the proposed method was used successfully in combination with Strang’s operator decomposition method.

Two low order characteristic finite element methods for a convection-dominated transport problem

Two low order rectangular finite elements for the convection–diffusion problem with a modified characteristic finite element scheme are studied in this paper. The O(h2) order error estimates in L2-norm with respect to the space are obtained for one element with regular meshes and the other under anisotropic meshes. In the process, we use some distinct properties of the interpolation operator, the integral identity technique and the mean value technique, instead of the traditional elliptic projection which is an indispensable tool in the convergence analysis of the previous literature. Finally, some numerical results are provided to verify our theoretical analysis.

A characteristics-mixed covolume method for a convection-dominated transport problem

In this paper, we propose a characteristics-mixed covolume method for approximating the solution to a convection dominated transport problem. The method is a combination of characteristic approximation to handle the convection term in time and mixed covolume method spatial approximation to deal with the diffusion term. The velocity and press are approximated by the lowest order Raviart–Thomas mixed finite element space on rectangles. The projection of a mixed covolume element is introduced. We prove its first order optimal rate of convergence for the approximate velocities in the L 2 norm as well as for the approximate pressures in the L 2 norm.

A low order anisotropic nonconforming characteristic finite element method for a convection-dominated transport problem

A low order anisotropic nonconforming rectangular finite element method for the convection–diffusion problem with a modified characteristic finite element scheme is studied in this paper. The O ( h 2 ) order error estimate in L 2 -norm with respect to the space, one order higher than the expanded characteristic-mixed finite element scheme with order O ( h ) , and the same as the conforming case for a modified characteristic finite element scheme under regular meshes, is obtained by use of some distinct properties of the interpolation operator and the mean value technique, instead of the so-called elliptic projection, which is an indispensable tool in the convergence analysis of the previous literature. Lastly, some numerical results of the element are provided to verify our theoretical analysis.

Explicit and implicit FEM-FCT algorithms with flux linearization

A new approach to the design of flux-corrected transport (FCT) algorithms for continuous (linear/multilinear) finite element approximations of convection-dominated transport problems is pursued. The algebraic flux correction paradigm is revisited, and a family of nonlinear high-resolution schemes based on Zalesak’s fully multidimensional flux limiter is considered. In order to reduce the cost of flux correction, the raw antidiffusive fluxes are linearized about an auxiliary solution computed by a high- or low-order scheme. By virtue of this linearization, the costly computation of solution-dependent correction factors is to be performed just once per time step, and there is no need for iterative defect correction if the governing equation is linear. A predictor–corrector algorithm is proposed as an alternative to the hybridization of high- and low-order fluxes. Three FEM-FCT schemes based on the Runge–Kutta, Crank–Nicolson, and backward Euler time-stepping are introduced. A detailed comparative study is performed for linear convection–diffusion equations.