Linear Convection Equation Research Articles

A transport-physics-accurate convection discretization based on first order upwind and the need for a new error assessment strategy

Several numerical schemes have been proposed in the literature to deal with convection phenomena, but all of them present some spurious effects. In this paper we show that, once a more physically-oriented error assessment tool is used in substitution for the classical wave number analysis, it is possible to develop an improved family of numerical schemes. Results are presented for a particular implementation based on first order upwind, in test cases involving linear and non-linear convective equation. This new scheme is able to simulate linear convection equation under uniform field with machine accuracy precision, with zero diffusion and dispersion. Moreover, the results for the non-linear convection are also significantly more accurate over the whole spectrum when compared to a traditional scheme based on a fifteen points stencil.

Accuracy-enhancement of discontinuous Galerkin solutions for convection-diffusion equations in multiple-dimensions

Discontinuous Galerkin (DG) methods exhibit “hidden accuracy” that makes superconvergence of this method an increasing popular topic to address. Previous investigations have focused on the superconvergent properties of ordinary differential equations and linear hyperbolic equations. Additionally, superconvergence of order k + 3 2 k+\frac {3}{2} for the convection-diffusion equation that focuses on a special projection using the upwind flux was presented by Cheng and Shu. In this paper we demonstrate that it is possible to extend the smoothness-increasing accuracy-conserving (SIAC) filter for use on the multi-dimensional linear convection-diffusion equation in order to obtain 2 k k + m m order of accuracy, where m m depends upon the flux and takes on the values 0 , 1 2 , 0,\, \frac {1}{2}, or 1. 1. The technique that we use to extract this hidden accuracy was initially introduced by Cockburn, Luskin, Shu, and Süli for linear hyperbolic equations and extended by Ryan et al. as a smoothness-increasing accuracy-conserving filter. We solve this convection-diffusion equation using the local discontinuous Galerkin (LDG) method and show theoretically that it is possible to obtain O ( h 2 k + m ) \mathcal {O}(h^{2k+m}) in the negative-order norm. By post-processing the LDG solution to a linear convection equation using a specially designed kernel such as the one by Cockburn et al., we can compute this same order accuracy in the L 2 L^2 -norm. Additionally, we present numerical studies that confirm that we can improve the LDG solution from O ( h k + 1 ) \mathcal {O}(h^{k+1}) to O ( h 2 k + 1 ) \mathcal {O}(h^{2k+1}) using alternating fluxes and that we actually obtain O ( h 2 k + 2 ) \mathcal {O}(h^{2k+2}) for diffusion-dominated problems.

An oscillation-free high order TVD/CBC-based upwind scheme for convection discretization

An oscillation-free high order scheme is presented for convection discretization by using the normalized-variable formulation in the finite volume framework. It adopts the cubic upwind interpolation scheme as the basic scheme so as to obtain high order accuracy in smooth solution domain. In order to avoid unphysical oscillations of numerical solutions, the present scheme is designed on the TVD (total variational diminishing) constraint and CBC (convection boundedness criterion) condition. Numerical results of several linear and nonlinear convection equations with smooth or discontinuous initial distributions demonstrate the present scheme possesses second-order accuracy, good robustness and high resolution.

Numerical resolution of discontinuous Galerkin methods for time dependent wave equations

The discontinuous Galerkin (DG) method is known to provide good wave resolution properties, especially for long time simulation. In this paper, using Fourier analysis, we provide a quantitative error analysis for the semi-discrete DG method applied to time dependent linear convection equations with periodic boundary conditions. We apply the same technique to show that the error is of order k + 2 superconvergent at Radau points on each element and of order 2 k + 1 superconvergent at the downwind point of each element, when using piecewise polynomials of degree k. An analysis of the fully discretized approximation is also provided. We compute the number of points per wavelength required to obtain a fixed error for several fully discrete schemes. Numerical results are provided to verify our error analysis.

Finite volume schemes for the approximation via characteristics of linear convection equations with irregular data

We consider the approximation by multidimensional finite volume schemes of the transport of an initial measure by a Lipschitz flow. We first consider a scheme defined via characteristics, and we prove the convergence to the continuous solution, as the time step and the ratio of the space step to the time step tend to zero. We then consider a second finite volume scheme, obtained from the first one by addition of some uniform numerical viscosity. We prove that this scheme converges to the continuous solution, as the space step tends to zero, whereas the ratio of the space step to the time step remains bounded by below and by above, and under assumption of uniform regularity of the mesh. This is obtained via an improved discrete Sobolev inequality and a sharp weak BV estimate, under some additional assumptions on the transport flow. Examples show the optimality of these assumptions.

Error bound between monotone difference schemes and their modified equations

It is widely believed that if monotone difference schemes are applied to the linear convection equation with discontinuous initial data, then solutions of the monotone schemes are closer to solutions of their parabolic modified equations than that of the original convection equation. We will confirm the conjecture in this paper. It is well known that solutions of the monotone schemes and their parabolic modified equations approach discontinuous solutions of the linear convection equation at a rate only half in the L 1 -norm. We will prove that the error bound between solutions of the monotone schemes and that of their modified equations is order one in the L 1 -norm. Therefore the conclusion shows that the monotone schemes solve the modified equations more accurately than the original convection equation even if the initial data is discontinuous. As a consequence of the main result, we will show that the half-order rate of convergence for the monotone schemes to the convection equation is the best possible.

GEOMETRIC INTEGRATION BY SOLUTION INTERPOLATION

Numerical schemes that are implemented by interpolation of exact solutions to a differential equation naturally preserve geometric properties of the differential equation. The solution interpolation method can be used for development of a new class of geometric integrators, which generally show better performances than standard method both quantitatively and qualitatively. Several examples including a linear convection equation and a nonlinear heat equation are included.

Composite high resolution localized relaxation scheme based on upwinding for hyperbolic conservation laws

AbstractIn this work we present an upwind‐based high resolution scheme using flux limiters. Based on the direction of flow we choose the smoothness parameter in such a way that it leads to a truly upwind scheme without losing total variation diminishing (TVD) property for hyperbolic linear systems where characteristic values can be of either sign. Here we present and justify the choice of smoothness parameters. The numerical flux function of a high resolution scheme is constructed using wave speed splitting so that it results into a scheme that truly respects the physical hyperbolicity property. Bounds are given for limiter functions to satisfy TVD property. The proposed scheme is extended for non‐linear problems by using the framework of relaxation system that converts a non‐linear conservation law into a system of linear convection equations with a non‐linear source term. The characteristic speed of relaxation system is chosen locally on three point stencil of grid. This obtained relaxation system is solved using composite scheme technique, i.e. using a combination of proposed scheme with the conservative non‐standard finite difference scheme. Presented numerical results show higher resolution near discontinuity without introducing spurious oscillations. Copyright © 2008 John Wiley & Sons, Ltd.

Time-reversibility of the Euler equations as a benchmark for energy conserving schemes

The three-dimensional incompressible Euler equations are time-reversible. This property should be preserved as well as possible by numerical discretizations. This article investigates the time-reversibility properties of various solvers designed for incompressible Navier–Stokes computations. The test case is the inviscid Taylor–Green vortex, which becomes “turbulent” before the time is reversed to try to recover the initial condition. The simulations are performed using high and low order finite difference solvers as well as using a pseudo-spectral solver. Various time-stepping schemes are also investigated. Although the flow statistics are significantly affected by the accuracy of the space discretization, the time-reversibility is not because most space-discretizations are time-reversible for an exact time-stepping. The crucial factor for time-reversibility is the accuracy of the time-stepping scheme and its interaction with the space-discretization. Furthermore, an important practical requirement for the solver is to be energy conserving in order to avoid numerical instability. An energy conserving solver using an accurate time-stepping is then able to go back almost perfectly from a complex “turbulent” flow to the simple initial condition. Therefore, we propose that this constitutes a severe and useful benchmark that Navier–Stokes solvers should challenge. The present investigations and their conclusions are also supported by parallel 1-D investigations, using the non-linear convection equation (inviscid Burgers) and the linear convection equation.

Modified equation for adaptive monotone difference schemes and its convergent analysis

A modified parabolic equation for adaptive monotone difference schemes based on equal-arclength mesh, applied to the linear convection equation, is derived and its convergence analysis shows that solutions of the modified equation approach a discontinuous (piecewise smooth) solution of the linear convection equation at order one rate in the L 1 -norm. It is well known that solutions of the monotone schemes with uniform meshes and their modified equation approach the same discontinuous solution at a half-order rate in the L 1 -norm. Therefore, the convergence analysis for the modified equation provided in this work demonstrates theoretically that the monotone schemes with adaptive grids can improve the solution accuracy. Numerical experiments also confirm the theoretical conclusions.

Improving spatial resolution characteristics of finite difference and finite volume schemes by approximate deconvolution pre-processing

A pre-processing step is proposed as a general method to enhance resolution properties of low order numerical differentiation and interpolation. Pre-processing operators are designed by taking two or more terms in the approximate deconvolution formula and using a local filter whose response characteristics are close to those of the numerical operation considered; operators for second order central differencing for first and second derivatives and also for a finite volume method are determined. In addition to the higher resolution the effective order of the truncation error can also be increased. The repeated filtering operations couple the operation over a wider stencil without using direct formulas. The effect of improving the resolution properties is illustrated first by computing the propagation of a square wave by a 1D, linear convection equation. Next, pre-processing is implemented in a standard finite volume code for solving Navier–Stokes equations for incompressible flow. In the Taylor–Green vortex flow, the improvements in order behaviour are demonstrated. The instability of plane Poiseuille flow, which is a sensitive test of resolution ability, shows that the predicted growth rates with the pre-processing scheme are more accurate than those obtained with a second order scheme. Direct numerical simulations of turbulent channel flow show that the improvement allows accurate solutions to be found on smaller grids.

Finite difference approximations of first derivatives for three-dimensional grid singularities

Explicit finite difference approximations of first derivatives are developed for three-dimensional four-block and six-block grid singularities. The schemes for the four-block singularity are of second, fourth and sixth order of accuracy, whereas the six-block schemes are of second and fourth order. The work extends a recently reported idea of developing schemes for two-dimensional grid singularities to three dimensions. Term matching of Taylor-series expansions of the dependent variable at neighboring points and the requirement for non-dissipative schemes do not provide enough equations to determine all coefficients; the open coefficients are determined through an optimization process to minimize the isotropy error of the numerical phase velocity. Owing to that optimization, the phase velocities and time-step limits for explicit time integration are very isotropic. The spectral characteristics, as dispersion and dissipation property and the time-step limit indicate that the proposed schemes for grid singularities can be well combined with Cartesian schemes of the same order of accuracy on the regular part of a computational mesh. The formal orders of accuracy of the proposed schemes are verified using the three-dimensional linear convection equation. Stability limits for explicit time integration are given.

Efficient high‐resolution relaxation schemes for hyperbolic systems of conservation laws

AbstractIn this work, we present a total variation diminishing (TVD) scheme in the zero relaxation limit for nonlinear hyperbolic conservation law using flux limiters within the framework of a relaxation system that converts a nonlinear conservation law into a system of linear convection equations with nonlinear source terms. We construct a numerical flux for space discretization of the obtained relaxation system and modify the definition of the smoothness parameter depending on the direction of the flow so that the scheme obeys the physical property of hyperbolicity. The advantages of the proposed scheme are that it can give second‐order accuracy everywhere without introducing oscillations for 1‐D problems (at least with) smooth initial condition. Also, the proposed scheme is more efficient as it works for any non‐zero constant value of the flux limiter ϕ ϵ [0, 1], where other TVD schemes fail. The resulting scheme is shown to be TVD in the zero relaxation limit for 1‐D scalar equations. Bound for the limiter function is obtained. Numerical results support the theoretical results. Copyright © 2007 John Wiley & Sons, Ltd.

The spatial resolution properties of composite compact finite differencing

Demand for spectral-like spatial routines to resolve fine-scale physics is easily satisfied by compact finite differencing. Commonly, the lower-order multi-parameter families at (and near) non-periodic boundaries are independently tuned to meet or exceed the high-order resolution character of the field stencil. Unfortunately, that approach quantifies a false influence of the boundary scheme on the resultant interior dispersive and dissipative consequences of the compound template. Knowing that each composite template owns three ingredients that define their numerical character, only their formal accuracy and global stability have been properly treated in a coupled fashion. The present work presents a companion means for quantifying the resultant spatial resolution properties. The procedure particularly focuses on the multi-parameter families used to diminish the dispersive and dissipative errors at the non-periodic boundaries. The process introduces a least-squares technique of the target field stencil to optimize the free parameters of the boundary scheme. Application of the optimized templates to both the linear convection and Burgers equations at a fictitious non-periodic boundary showed major reductions of the predictive error.

A grid‐free upwind relaxation scheme for inviscid compressible flows

AbstractA new grid‐free upwind relaxation scheme for simulating inviscid compressible flows is presented in this paper. The non‐linear conservation equations are converted to linear convection equations with non‐linear source terms by using a relaxation system and its interpretation as a discrete Boltzmann equation. A splitting method is used to separate the convection and relaxation parts. Least squares upwinding is used for discretizing the convection equations, thus developing a grid‐free scheme which can operate on any arbitrary distribution of points. The scheme is grid free in the sense that it works on any arbitrary distribution of points and it does not require any topological information like elements, faces, edges, etc. This method is tested on some standard test cases. To explore the power of the grid‐free scheme, solution‐based adaptation of points is done and the results are presented, which demonstrate the efficiency of the new grid‐free scheme. Copyright © 2005 John Wiley & Sons, Ltd.

Pointwise Green's function estimates toward stability for multidimensional fourth-order viscous shock fronts

For the case of multidimensional viscous conservation laws with fourth-order smoothing only, we develop detailed pointwise estimates on the Green's function for the linear fourth-order convection equation that arises upon linearization of the conservation law about a viscous planar wave solution. As in previous analyses in the case of second-order smoothing, our estimates are sufficient to establish that spectral stability implies nonlinear stability, though the full development of this result will be considered in a companion paper.

High Accuracy Compact Schemes and Gibbs' Phenomenon

Compact difference schemes have been investigated for their ability to capture discontinuities. A new proposed scheme (Sengupta, Ganerwal and De (2003). J. Comp. Phys. 192(2), 677.) is compared with another from the literature Zhong (1998). J. Comp. Phys. 144, 622 that was developed for hypersonic transitional flows for their property related to spectral resolution and numerical stability. Solution of the linear convection equation is obtained that requires capturing discontinuities. We have also studied the performance of the new scheme in capturing discontinuous solution for the Burgers equation. A very simple but an effective method is proposed here in early diagnosis for evanescent discontinuities. At the discontinuity, we switch to a third order one-sided stencil, thereby retaining the high accuracy of solution. This produces solution with vastly reduced Gibbs' phenomenon of the solution. The essential causes behind Gibbs' phenomenon is also explained.

On unconditional conservation of kinetic energy by finite-difference discretizations of the linear and non-linear convection equation

In the past the development of kinetic energy conserving finite-difference methods mostly focused on second-order accurate central methods defined on uniform grids. Nowadays the need for high-order accurate discretizations, to perform for instance accurate numerical simulations of turbulent flow, calls for the development of novel kinetic energy conserving discretization schemes. Instead of choosing a fixed basis discretization up front, in this paper a different, more general, approach is applied. For a Cartesian mesh, sets of conditions are presented such that all discretizations of the linear or non-linear convection equation which obey these conditions, unconditionally conserve kinetic energy. For the linear convection equation it is shown that on a uniform grid it is necessary and sufficient for a discretization to be central in order to be fully conservative, that is: such discretizations not only unconditionally conserve kinetic energy but also unconditionally conserve momentum. On non-uniform grids an algorithm is introduced that can be used to generate fully conservative discretizations that are at least first-order accurate. The derivation of the discretization conditions for the non-linear convection equation is performed in the two-dimensional (2D) linear case. Some examples on uniform grids and on non-uniform grids are presented. It is shown that on uniform grids no upper limit exists with respect to the accuracy of the kinetic energy conserving method. For the higher-dimensional linear and non-linear convection equation the same set of conditions, which ensure the unconditional conservation of kinetic energy, are found as in the 2D linear case. Other results too are found to be straightforward generalizations of the corresponding 2D linear results. It is shown that the fourth-order unconditionally kinetic energy conserving discretization on a staggered mesh introduced in this paper is well suited to simulate the initial development of an inviscid shear layer instability in a divergence-free flow.

Space—time CESE method for the convection equation

In the paper, we study the analytic convergence of the space-time conservation element and solution element method for smooth solutions of the convection equation. Stability of the method is limited by the CFL condition and a parameter,e,which controls the numerical dissipation. We show that the method converges under appropriate conditions on the mesh and on the parameter,e, by considering a one-dimensional convection equation. The main advantage of this second order scheme is that it is an explicit marching scheme that allows one to solve for both the function and its derivative at the same time with comparable accuracy.Numerical simulations are presented to verify the convergence results for both the linear convection equation and a non-linear conservation law.

Membrane diffusion and flux response

A linear diffusion convection equation for the transport of a single chemical component through a membrane is coupled with a linear reaction equation describing interaction of the diffusing chemical with an immobile reactant. Expressions are derived for the flux responses in and out of the membrane to given concentrations applied at one side of the membrane while zero concentration is maintained at the other. These fluxes are compared with the corresponding fluxes for the complementary experiment in which the imposed boundary concentration at the two faces of the membrane are interchanged. Ussing-type flux-ratio relationships are obtained for the outflow fluxes, but the inflow fluxes respond in a different way, and it is found that the difference of the influxes for complementary experiments remain constant for all time.