Convex ENO Schemes for Hamilton–Jacobi Equations

In one dimension, viscosity solutions of Hamilton–Jacobi (HJ) equations can be thought as primitives of entropy solutions for conservation laws. Based on this idea, both theoretical and numerical concepts used for conservation laws can be passed to HJ equations even in several dimensions. In this paper, we construct convex ENO (CENO) schemes for HJ equations. This construction is a generalization from the work by Liu and Osher on CENO schemes for conservation laws. Several numerical experiments are performed. L ¹ and L ^∞ error and convergence rate are calculated as well.     

A weighted ENO-flux limiter scheme for hyperbolic conservation laws

We present a new scheme that combines essentially non-oscillatory (ENO) reconstructions together with monotone upwind schemes for scalar conservation laws interpolants. We modify a second-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. Numerical experiments are done in order to compare a weighted version of the hybrid scheme to weighted essentially non-oscillatory (WENO) schemes with constant Courant–Friedrichs–Lewy number under relaxed step size restrictions. Our results show that the new scheme reduces smearing near shocks and corners, and in some cases it is more accurate near discontinuities compared with higher-order WENO schemes. The hybrid scheme avoids spurious oscillations while using a simple componentwise extension for solving hyperbolic systems. The new scheme is less damped than WENO schemes of comparable accuracy and less oscillatory than higher-order WENO schemes. Further experiments are done on multi-dimensional problems to show that our scheme remains non-oscillatory while giving good resolution of discontinuities.     

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.     

Shock Capturing Artificial Dissipation for High-Order Finite Difference Schemes

We study 2nd-, 4th-, 6th- and 8th-order accurate finite difference schemes approximating systems of conservation laws. Our goal is to utilize the high order of accuracy of the schemes for approximating complicated flow structures and add suitable diffusion operators to capture shocks. We choose appropriate viscosity terms and prove non-linear entropy stability. In the scalar case, entropy stability enables us to prove convergence to the unique entropy solution. Moreover, a limiter function that localizes the effect of the dissipation around discontinuities is derived. The resulting scheme is entropy stable for systems, and also converges to the entropy solution in the scalar case. We present a number of numerical experiments in order to demonstrate the robustness and accuracy of our scheme. The set of examples consists of a moving shock solution to the Burgers’ equation, a solution to the Euler equations that consists of a rarefaction and two contact discontinuities and a shock/entropy wave solution to the Euler equations (Shu’s test problem). Furthermore, we use the limited scheme to compute the solution to the linear advection equation and demonstrate that the limiter quickly vanishes for smooth flows and design/high-order of accuracy is retained. The numerical results in all experiments were very good. We observe a remarkable gain in accuracy when the order of the scheme is increased.     

二维双曲守恒律标量方程的三阶CWENO-型熵相容算法

应用提出的中心加权基本无振荡(CWENO)-型熵相容格式求解了二维双曲守恒律方程初边值问题,对所得数值结果进行了分析与讨论,并通过与准确解的比较发现该数值求解格式稳定性条件可以取到0.6,而激波过渡带只有1~2个网格单元。实验结果表明该数值求解格式分辨率高且数值稳定性好。     

Entropy-TVD Scheme for the Shallow Water Equations in One Dimension

The Entropy-TVD scheme was developed for the non-linear scalar conservation laws in Chen and Mao (J Sci Comput 47:150–169, 2011). The scheme with step reconstruction simultaneously computes the two numerical entities, the numerical solution and the numerical entropy, and numerical examples show that the scheme provides a super-convergence rate. In this paper, we extend an Entropy-TVD scheme to the shallow water equations in one dimension. We prove that the scheme satisfies the entropy condition. Numerical tests show that the Entropy-TVD scheme has better resolution than the standard Godunov scheme.     

A New Smoothness Indicator for the WENO Schemes and Its Effect on the Convergence to Steady State Solutions

The convergence to steady state solutions of the Euler equations for the fifth-order weighted essentially non-oscillatory (WENO) finite difference scheme with the Lax–Friedrichs flux splitting [7, (1996) J. Comput. Phys. 126, 202–228.] is studied through systematic numerical tests. Numerical evidence indicates that this type of WENO scheme suffers from slight post-shock oscillations. Even though these oscillations are small in magnitude and do not affect the “essentially non-oscillatory” property of WENO schemes, they are indeed responsible for the numerical residue to hang at the truncation error level of the scheme instead of settling down to machine zero. We propose a new smoothness indicator for the WENO schemes in steady state calculations, which performs better near the steady shock region than the original smoothness indicator in [7, (1996) J. Comput. Phys. 126, 202–228.]. With our new smoothness indicator, the slight post-shock oscillations are either removed or significantly reduced and convergence is improved significantly. Numerical experiments show that the residue for the WENO scheme with this new smoothness indicator can converge to machine zero for one and two dimensional (2D) steady problems with strong shock waves when there are no shocks passing through the domain boundaries. Dedicated to the memory of Professor Xu-Dong Liu.     

求解双曲守恒律及其对流扩散方程的中心迎风方法

提出了一种新的求解双曲守恒律方程(组)的四阶半离散中心迎风差分方法．空间导数项的离散采用四阶CWENO(central weighted essentially non—oscillatory)的构造方法,使所得到的新方法在提高精度的同时,具有更高的分辨率．使用该方法产生的数值粘性要比交错的中心格式小,而且由于数值粘性与时间步长无关,从而时间步长可根据稳定性需要尽可能的小．     

Mesh Redistribution Strategies and Finite Element Schemes for Hyperbolic Conservation Laws

In this work we consider a new class of Relaxation Finite Element schemes for hyperbolic conservation laws, with more stable behavior on the limit area of the relaxation parameter. Combining this scheme with an efficient adapted spatial redistribution process considered also in this work, we form a robust scheme of controllable resolution. The results on a number of test problems show that this scheme can produce entropic-approximations of high resolution, even on the limit of the relaxation parameter where the scheme lacks of the relaxation mechanism. Thus we experimentally conclude that the proposed spatial redistribution process, has by its own interesting stabilization properties for computational solutions of conservation law problems.     

A high-order non-oscillatory central scheme with non-staggered grids for hyperbolic conservation laws

In this work, we present a scheme which is based on non-staggered grids. This scheme is a new family of non-staggered central schemes for hyperbolic conservation laws. Motivation of this work is a staggered central scheme recently introduced by A.A.I. Peer et al. [A new fourth-order non-oscillatory central scheme for hyperbolic conservation laws, Appl. Numer. Math. 58 (2008) 674–688]. The most important properties of the technique developed in the current paper are simplicity, high-resolution and avoiding the use of staggered grids and hence is simpler to implement in frameworks which involve complex geometries and boundary conditions. Numerical implementation of the new scheme is carried out on the scalar conservation laws with linear, non-linear flux and systems of hyperbolic conservation laws. The numerical results confirm the expected accuracy and high-resolution properties of the scheme.     

Solutions of Multi-dimensional Hyperbolic Systems of Conservation Laws by Square Entropy Condition Satisfying Discontinuous Galerkin Method

In this paper, we study formally high-order accurate discontinuous Galerkin methods on general arbitrary grid for multi-dimensional hyperbolic systems of conservation laws [Cockburn, B., and Shu, C.-W. (1989, Math. Comput. 52, 411–435, 1998, J. Comput. Phys. 141, 199–224); Cockburn et al. (1989, J. Comput. Phys. 84, 90–113; 1990, Math. Comput. 54, 545–581). We extend the notion of E-flux [Osher (1985) SIAM J. Numer. Anal. 22, 947–961] from scalar to system, and found that after flux splitting upwind flux [Cockburn et al. (1989) J. Comput. Phys. 84, 90–113] is a Riemann solver free E-flux for systems. Therefore, we are able to show that the discontinuous Galerkin methods satisfy a cell entropy inequality for square entropy (in semidiscrete sense) if the multi-dimensional systems are symmetric. Similar result [Jiang and Shu (1994) Math. Comput. 62, 531–538] was obtained for scalar equations in multi-dimensions. We also developed a second-order finite difference version of the discontinuous Galerkin methods. Numerical experiments have been obtained with excellent results.      

Comparing Notions of Computational Entropy

In the information theoretic world, entropy is both the measure of randomness in a source and a bound for the compression achievable for that source by any encoding scheme. But when we must restrict ourselves to efficient schemes, entropy no longer captures these notions well. For example, there are distributions with very low entropy that nonetheless look random for polynomial-bound algorithms. Different notions of computational entropy have been proposed to take the role of entropy in such settings. Results in Goldberg and Sipser (SIAM J. Comput. 20(3):524–536, 1991) and Wee (IEEE conference on computational complexity, pp. 29–41, 2004) suggest that when time bounds are introduced, the entropy of a distribution no longer coincides with the most effective compression for that source. This paper analyses three measures that try to capture the compressibility of a source, establishing relations and separations between them and analysing the two special cases of the uniform and the universal distribution m ^t over binary strings of a fixed size. It is shown that for the uniform distribution the three measures are equivalent and that for m ^t there is a clear separation between metric type entropy and the maximum compressibility of a source. Partially supported by KCrypt (POSC/EIA/60819/2004), the grant SFRH/BD/13124/2003 from FCT and funds granted to LIACC through the Programa de Financiamento Plurianual, FCT and Programa POSI.     

Finite Volume HWENO Schemes for Nonconvex Conservation Laws

Following the previous work of Qiu and Shu (SIAM J Sci Comput 31: 584–607, 2008), we investigate the performance of Hermite weighted essentially non-oscillatory (HWENO) scheme for nonconvex conservation laws. Similar to many other high order methods, we show that the finite volume HWENO scheme performs poorly for some nonconvex conservation laws. We modify the scheme around the nonconvex regions, based on a first order monotone scheme and a second entropic projection, to ensure entropic convergence. Extensive numerical tests are performed. Compare with the earlier work of Qiu and Shu which focuses on 1D scalar problems, we apply the modified schemes (both WENO and HWENO) to one-dimensional Euler system with nonconvex equation of state and two-dimensional problems.     

Local discontinuous Galerkin methods based on the multisymplectic formulation for two kinds of Hamiltonian PDEs

ABSTRACT

This paper examines the novel local discontinuous Galerkin (LDG) discretization for Hamiltonian PDEs based on its multisymplectic formulation. This new kind of LDG discretizations possess one major advantage over other standard LDG method, which, through specially chosen numerical fluxes, states the preservation of discrete conservation laws (i.e. energy), and also the multisymplectic structure while the symplectic time integration is adopted. Moreover, the corresponding local multisymplectic conservation law holds at the units of elements instead of each node. Taking the nonlinear Schrödinger equation and the KdV equation as the examples, we illustrate the derivations of discrete conservation laws and the corresponding numerical fluxes. Numerical experiments by using the modified LDG method are demonstrated for the sake of validating our theoretical results.     

A Local Extrapolation Method for Hyperbolic Conservation Laws. I. The ENO Underlying Schemes

In this paper, we introduce a local extrapolation method (LEM) for the essentially non-oscillatory (ENO) schemes solving nonlinear hyperbolic conservation laws. The method extrapolates the numerical flux of the underlying scheme so that it keeps conservativity. We use a minmod type limiter to avoid spurious oscillations. We propose a new balancing technique that preserves the symmetry of a symmetric wave that works well for a wide range of CFL numbers. We also introduce two artificial compression procedures to the LEM which yield sharp resolutions of contact discontinuities. Numerical examples are presented to illustrate the performance of the method.     

Convection of Concentrated Vortices and Passive Scalars as Solitary Waves

A new version of a computational method, Vorticity Confinement, is described. Vorticity Confinement has been shown to efficiently treat thin features in multi-dimensional incompressible fluid flow, such as vortices and streams of passive scalars, and to convect them over long distances with no spreading due to numerical errors. Outside the features, where the flow is irrotational or the scalar vanishes, the method automatically reduces to conventional discretized finite difference fluid dynamic equations. The features are treated as a type of weak solution and, within the features, a nonlinear difference equation, as opposed to finite difference equation, is solved that does not necessarily represent a Taylor expansion discretization of a simple partial differential equation (PDE). The approach is similar to artificial compression and shock capturing schemes, where conservation laws are satisfied across discontinuities. For the features, the result of this conservation is that integral quantities such as total amplitude and centroid motion are accurately computed. Basically, the features are treated as multi-dimensional nonlinear discrete solitary waves that live on the computational lattice. These obey a confinement relation that is a generalization to multiple dimensions of 1-D discontinuity capturing schemes. A major point is that the method involves a discretization of a rotationally invariant operator, rather than a composition of separate 1-D operators, as in conventional discontinuity capturing schemes. The main objective of this paper is to introduce a new formulation of Vorticity Confinement that, compared to the original formulation, is simpler, allows more detailed analysis, and exactly conserves momentum for vortical flow. First, a short critique of conventional methods for these problems is given. The basic new method is then described. Finally, analysis of the new method and initial results are presented.     

Conservative and Stable Degree Preserving SBP Operators for Non-conforming Meshes

Non-conforming numerical approximations offer increased flexibility for applications that require high resolution in a localized area of the computational domain or near complex geometries. Two key properties for non-conforming methods to be applicable to real world applications are conservation and energy stability. The summation-by-parts (SBP) property, which certain finite-difference and discontinuous Galerkin methods have, finds success for the numerical approximation of hyperbolic conservation laws, because the proofs of energy stability and conservation can discretely mimic the continuous analysis of partial differential equations. In addition, SBP methods can be developed with high-order accuracy, which is useful for simulations that contain multiple spatial and temporal scales. However, existing non-conforming SBP schemes result in a reduction of the overall degree of the scheme, which leads to a reduction in the order of the solution error. This loss of degree is due to the particular interface coupling through a simultaneous-approximation-term (SAT). We present in this work a novel class of SBP–SAT operators that maintain conservation, energy stability, and have no loss of the degree of the scheme for non-conforming approximations. The new degree preserving discretizations require an ansatz that the norm matrix of the SBP operator is of a degree \(\ge 2p\), in contrast to, for example, existing finite difference SBP operators, where the norm matrix is \(2p-1\) accurate. We demonstrate the fundamental properties of the new scheme with rigorous mathematical analysis as well as numerical verification.