Schemes For Scalar Conservation Laws Research Articles

Third order maximum-principle-satisfying and positivity-preserving Lax-Wendroff discontinuous Galerkin methods for hyperbolic conservation laws

There have been intensive studies on maximum-principle-satisfying and positivity-preserving methods for hyperbolic conservation laws. Most of them are based on the method of lines type time marching approaches, e.g. the Runge-Kutta methods, multi-step methods and backward Euler method. As an alternative, the Lax-Wendroff time marching approach utilizes the information of PDEs in the Taylor expansion of the solution in time, hence it is a high order and single-stage method. In this work, we propose third order maximum-principle-satisfying and positivity-preserving schemes for scalar conservation laws and the Euler equations based on the Lax-Wendroff time discretization and discontinuous Galerkin spatial discretization. The accuracy and effectiveness of the maximum-principle-satisfying and positivity-preserving techniques are demonstrated by ample numerical tests.

Maximum principle and positivity-preserving high order spectral volume schemes with parametrized flux limiters for solving hyperbolic conservation laws

In this paper, we use maximum principle preserving (MPP) and positivity-preserving (PP) parametrized flux limiters to achieve strict maximum principle and positivity-preserving property for the high order spectral volume (SV) scheme for solving hyperbolic conservation laws. This research is based on a generalization of the MPP and PP parametrized flux limiters in Xu (2014) and Christlieb et al. (2015) with Runge–Kutta (RK) time discretizations. For constructing MPP (PP) RK-SV schemes for hyperbolic conservation laws, we first focus on the RK-SV schemes to discuss how to apply MPP or PP parametrized flux limiters. Then we design and analyze high order MPP RK-SV schemes for scalar conservation laws, and high order PP RK-SV schemes for compressible Euler systems. The efficiency and effectiveness of the proposed schemes are demonstrated via a set of numerical experiments. Both the analysis and numerical experiments indicate that the proposed scheme without any additional time step restriction, not only preserves the maximum principle of the numerical approximation, but also maintains the designed high-order accuracy of the SV scheme for linear advection problems.

On the range diminishing property of numerical schemes for scalar conservation laws

Exact solutions of scalar conservation laws have the property that maxima are non-increasing, minima are non-decreasing and no new extrema are created. We present a simple and reliable general condition for numerical schemes to mimic this property. We demonstrate the result by applying it to various schemes, which leads to various CFL-like conditions.

A new modified Local Lax–Friedrichs scheme for scalar conservation laws with discontinuous flux

The interface flux plays an important role in designing numerical methods for scalar conservation laws with discontinuous flux function in space. Based on the (A,B)-type entropy solutions defined by Bürger et al. (2009), we introduce a new modified Local Lax–Friedrichs interface flux to approximate this system. The interface flux is Lipschitz continuous and monotone with respect to its two variables. Moreover, we present some numerical experiments to demonstrate the efficiency of our method.

Convergence Rates of Monotone Schemes for Conservation Laws with Discontinuous Flux

We prove that a class of monotone finite volume schemes for scalar conservation laws with discontinuous flux converge at a rate of $\sqrt{\Delta x}$ in $\mathrm{L}^1$, whenever the flux is strictly monotone in $u$ and the spatial dependency of the flux is piecewise constant with finitely many discontinuities. We also present numerical experiments to illustrate the main result. To the best of our knowledge, this is the first proof of any type of convergence rate for numerical methods for conservation laws with discontinuous, nonlinear flux. Our proof relies on convergence rates for conservation laws with initial and boundary value data. Since those are not readily available in the literature we establish convergence rates in that case en passant in the Appendix.

Convergence of the follow-the-leader scheme for scalar conservation laws with space dependent flux

This paper deals with the derivation of entropy solutions to Cauchy problems for a class of scalar conservation laws with space-density depending fluxes from systems of deterministic particles of follow-the-leader type. We consider fluxes which are product of a function of the density \begin{document}$ v(\rho) $\end{document} and a function of the space variable \begin{document}$ \phi(x) $\end{document} . We cover four distinct cases in terms of the sign of \begin{document}$ \phi $\end{document} , including cases in which the latter is not constant. The convergence result relies on a local maximum principle and on a uniform \begin{document}$ BV $\end{document} estimate for the approximating density.

Flux-corrected transport algorithms preserving the eigenvalue range of symmetric tensor quantities

This paper presents a new approach to constraining the eigenvalue range of symmetric tensors in numerical advection schemes based on the flux-corrected transport (FCT) algorithm and a continuous finite element discretization. In the context of element-based FEM-FCT schemes for scalar conservation laws, the numerical solution is evolved using local extremum diminishing (LED) antidiffusive corrections of a low order approximation which is assumed to satisfy the relevant inequality constraints. The application of a limiter to antidiffusive element contributions guarantees that the corrected solution remains bounded by the local maxima and minima of the low order predictor.The FCT algorithm to be presented in this paper guarantees the LED property for the maximal and minimal eigenvalues of the transported tensor at the low order evolution step. At the antidiffusive correction step, this property is preserved by limiting the antidiffusive element contributions to all components of the tensor in a synchronized manner. The definition of the element-based correction factors for FCT is based on perturbation bounds for auxiliary tensors which are constrained to be positive semidefinite to enforce the generalized LED condition. The derivation of sharp bounds involves calculating the roots of polynomials of degree up to 3. As inexpensive and numerically stable alternatives, limiting techniques based on appropriate estimates are considered. The ability of the new limiters to enforce local bounds for the eigenvalue range is confirmed by numerical results for 2D advection problems.

On the convergence of a local third order shock capturing method for hyperbolic conservation laws

The Piecewise Polynomial Harmonic Method (PPHM) is a local third order accurate shock capturing method for hyperbolic conservation laws. In this paper, theoretical stability properties are presented. Using these properties, the convergence of the scheme for scalar conservation laws is obtained. A direct adaptation of these results should be used to derive the convergence of the PHM (Marquina, SIAM J. Sci. Comput. 15(4), 892---915, 1994). Finally, the method is tested in several classical problems in order to explore its numerical behavior.

On the Convergence of Space-Time Discontinuous Galerkin Schemes for Scalar Conservation Laws

We prove convergence of a class of space-time discontinuous Galerkin schemes for scalar hyperbolic conservation laws. Convergence to the unique entropy solution is shown for all orders of polynomial approximation, provided strictly monotone flux functions and a suitable shock-capturing operator are used. The main improvement, compared to previously published results of similar scope, is that no streamline-diffusion stabilization is used. This is the way discontinuous Galerkin schemes were originally proposed, and are most often used in practice.

Second-Order Convergence of Monotone Schemes for Conservation Laws

We prove that a class of monotone, \emph{$W_1$-contractive} schemes for scalar conservation laws converge at a rate of $\Delta x^2$ in the Wasserstein distance ($W_1$-distance), whenever the initial data is decreasing and consists of a finite number of piecewise constants. It is shown that the Lax--Friedrichs, Enquist--Osher and Godunov schemes are $W_1$-contractive. Numerical experiments are presented to illustrate the main result. To the best of our knowledge, this is the first proof of second-order convergence of any numerical method for discontinuous solutions of nonlinear conservation laws.

High-order accurate, fully discrete entropy stable schemes for scalar conservation laws

The recently developed TECNO schemes for hyperbolic conservation laws are designed to be high-order accurate and entropy stable, but are, as of yet, only semidiscrete. We perform an explicit discretization of the temporal derivative to obtain a fully discrete scheme, and derive a rather unrestrictive CFL condition that ensures global entropy stability. The scheme is tested in a series of numerical experiments.

A multiple-relaxation-time lattice Boltzmann method for high-speed compressible flows**Project supported by the Innovation Fund for Aerospace Science and Technology of China (Grant No. 2009200066) and the Aeronautical Science Fund of China (Grant No. 20111453012).

This paper presents a coupling compressible model of the lattice Boltzmann method. In this model, the multiple-relaxation-time lattice Boltzmann scheme is used for the evolution of density distribution functions, whereas the modified single-relaxation-time (SRT) lattice Boltzmann scheme is applied for the evolution of potential energy distribution functions. The governing equations are discretized with the third-order Monotone Upwind Schemes for scalar conservation laws finite volume scheme. The choice of relaxation coefficients is discussed simply. Through the numerical simulations, it is found that compressible flows with strong shocks can be well simulated by present model. The numerical results agree well with the reference results and are better than that of the SRT version.

A lattice Boltzmann model for simulation of compressible flows

SummaryThis paper presents a new model of lattice Boltzmann method for full compressible flows. On the basis of multi‐speed model, an extra potential energy distribution function is introduced to recover the full compressible Navier–Stokes equations with a flexible specific‐heat ratio and Prandtl number. The Chapman–Enskog expansion of the kinetic equations is performed, and the two‐dimension‐seventeen‐velocity density equilibrium distribution functions are obtained. The governing equations are discretized using the third order monotone upwind scheme for scalar conservation laws finite volume scheme. The van Albada limiter is used to avoid spurious oscillations. In order to verify the accuracy of this double‐distribution‐function model, the Riemann problems, Couette flows, and flows around a NACA0012 airfoil are simulated. It is found that the proposed lattice Boltzmann model is suitable for compressible flows, even for strong shock wave problem, which has an extremely large pressure ratio, 100,000. Copyright © 2014 John Wiley & Sons, Ltd.

A Maximum-Principle-Satisfying High-Order Finite Volume Compact WENO Scheme for Scalar Conservation Laws with Applications in Incompressible Flows

In this paper, a maximum-principle-satisfying finite volume compact scheme is proposed for solving scalar hyperbolic conservation laws. The scheme combines weighted essentially non-oscillatory schemes (WENO) with a class of compact schemes under a finite volume framework, in which the nonlinear WENO weights are coupled with lower order compact stencils. The maximum-principle-satisfying polynomial rescaling limiter in Zhang and Shu (J Comput Phys 229:3091---3120, 2010, Proc R Soc A Math Phys Eng Sci 467:2752---2776, 2011) is adopted to construct the present schemes at each stage of an explicit Runge---Kutta method, without destroying high order accuracy and conservativity. Numerical examples for one and two dimensional problems including incompressible flows are presented to assess the good performance, maximum principle preserving, essentially non-oscillatory and high resolution of the proposed method.

High order parametrized maximum-principle-preserving and positivity-preserving WENO schemes on unstructured meshes

In this paper, we generalize the maximum-principle-preserving (MPP) flux limiting technique developed by Xu (2013) [20] to a class of high order finite volume weighted essentially non-oscillatory (WENO) schemes for scalar conservation laws and the compressible Euler system on unstructured meshes in one and two dimensions. The key idea of this parameterized limiting technique is to limit the high order numerical flux with a first order flux which preserves the MPP or positivity-preserving (PP) property. The main purpose of this paper is to investigate the flux limiting approach with high order finite volume method on unstructured meshes which are often needed for solving some important problems on irregular domains. Truncation error analysis based on one-dimensional nonuniform meshes is presented to justify that the proposed MPP schemes can maintain third order accuracy in space and time. We also demonstrate through smooth test problems that the proposed third order MPP/PP WENO schemes coupled with a third order Runge–Kutta (RK) method attain the desired order of accuracy. Several test problems containing strong shocks and complex domain geometries are also presented to assess the performance of the schemes.

Second order scheme for scalar conservation laws with discontinuous flux

Burger, Karlsen, Torres and Towers in [9] proposed a flux TVD (FTVD) second order scheme with Engquist–Osher flux, by using a new nonlocal limiter algorithm for scalar conservation laws with discontinuous flux modeling clarifier thickener units. In this work we show that their idea can be used to construct FTVD second order scheme for general fluxes like Godunov, Engquist–Osher, Lax–Friedrich,… satisfying (A, B)-interface entropy condition for a scalar conservation law with discontinuous flux with proper modification at the interface. Also corresponding convergence analysis is shown. We show further from numerical experiments that solutions obtained from these schemes are comparable with the second order schemes obtained from the minimod limiter.

High order accurate and low dissipation method for unsteady compressible viscous flow computation on helicopter rotor in forward flight

An improved fifth-order weighted essentially non-oscillatory (WENO-Z) scheme combined with the moving overset grid technique has been developed to compute unsteady compressible viscous flows on the helicopter rotor in forward flight. In order to enforce periodic rotation and pitching of the rotor and relative motion between rotor blades, the moving overset grid technique is extended, where a special judgement standard is presented near the odd surface of the blade grid during search donor cells by using the Inverse Map method. The WENO-Z scheme is adopted for reconstructing left and right state values with the Roe Riemann solver updating the inviscid fluxes and compared with the monotone upwind scheme for scalar conservation laws (MUSCL) and the classical WENO scheme. Since the WENO schemes require a six point stencil to build the fifth-order flux, the method of three layers of fringes for hole boundaries and artificial external boundaries is proposed to carry out flow information exchange between chimera grids. The time advance on the unsteady solution is performed by the full implicit dual time stepping method with Newton type LU-SGS subiteration, where the solutions of pseudo steady computation are as the initial fields of the unsteady flow computation. Numerical results on non-variable pitch rotor and periodic variable pitch rotor in forward flight reveal that the approach can effectively capture vortex wake with low dissipation and reach periodic solutions very soon.

A high-order symmetrical weighted hybrid ENO-flux limiter scheme for hyperbolic conservation laws

In this paper, we propose a new weighted essentially non-oscillatory (WENO) procedure for solving hyperbolic conservation laws, on uniform meshes. The new scheme combines essentially non-oscillatory (ENO) reconstructions together with monotone upwind schemes for scalar conservation laws’ interpolants. In a one-dimensional context, first, we obtain an optimum polynomial on a five-cells stencil. This optimum polynomial is fifth-order accurate in regions of smoothness. Next, we modify a third-order ENO polynomial by choosing an additional point inside the stencil in order to obtain the highest accuracy when combined with the Harten–Osher reconstruction–evolution method limiter. Then, we consider the optimum polynomial as a symmetric and convex combination of four polynomials with ideal weights. After that, following the methodology of the classic WENO procedure, we calculate non-oscillatory weights with the ideal weights. Also, the numerical solution is advanced in time by means of the linear multi-step total variation bounded (TV B0) technique. Numerical examples on both scalar and gas dynamics problems confirm that the new scheme is non-oscillatory and yields sharp results when solving profiles with discontinuities. Comparing the new scheme with high-order WENO schemes shows that our method reduces smearing near shocks and corners, and in some cases it is more accurate near discontinuities. Finally, the new method is extended to multi-dimensional problems by a dimension-by-dimension approach. Several multi-dimensional examples are performed to show that our method remains non-oscillatory while giving good resolution of discontinuities.

Monotone first‐order weighted schemes for scalar conservation laws

AbstractIn this work, we present a monotone first‐order weighted (FORWE) method for scalar conservation laws using a variational formulation. We prove theoretical properties as consistency, monotonicity, and convergence of the proposed scheme for the one‐dimensional (1D) Cauchy problem. These convergence results are extended to multidimensional scalar conservation laws by a dimensional splitting technique. For the validation of the FORWE method, we consider some standard bench‐mark tests of bidimensional and 1D conservation law equations. Finally, we analyze the accuracy of the method with L1 and L∞ error estimates. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

A semi-discrete central scheme for scalar hyperbolic conservation laws with heterogeneous storage coefficient and its application to porous media flow

SUMMARY In this paper, we develop a new Godunov-type semi-discrete central scheme for a scalar conservation law on the basis of a generalization of the Kurganov and Tadmor scheme, which allows for spatial variability of the storage coefficient (e.g. porosity in multiphase flow in porous media) approximated by piecewise constant interpolation. We construct a generalized numerical flux at element edges on the basis of a nonstaggered inhomogeneous dual mesh, which reproduces the one postulated by Kurganov and Tadmor under the assumption of homogeneous storage coefficient. Numerical simulations of two-phase flow in strongly heterogeneous porous media illustrate the performance of the proposed scheme and highlight the important rule of the permeability–porosity correlation on finger growth and breakthrough curves. Copyright © 2013 John Wiley & Sons, Ltd.