In this paper, we develop a new sixth-order WENO scheme by adopting a convex combination of a sixth-order global reconstruction and four low-order local reconstructions. Unlike the classical WENO schemes, the associated linear weights of the new scheme can be any positive numbers with the only requirement that their sum equals one. Further, a very simple smoothness indicator for the global stencil is proposed. The new scheme can achieve sixth-order accuracy in smooth regions. Numerical tests in some one- and two-dimensional benchmark problems show that the new scheme has a little bit higher resolution compared with the recently developed sixth-order WENO-Z6 scheme, and it is more efficient than the classical fifth-order WENO-JS5 scheme and the recently developed sixth-order WENO6-S scheme.
In this paper, we apply high-order finite difference (FD) schemes for multispecies and multireaction detonations (MMD). In MMD, the density and pressure are positive and the mass fraction of the ith species in the chemical reaction, say $z_i$, is between 0 and 1, with $\sum z_i=1$. Due to the lack of maximum-principle, most of the previous bound-preserving technique cannot be applied directly. To preserve those bounds, we will use the positivity-preserving technique to all the $z_i'\text{s}$ and enforce $\sum z_i=1$ by constructing conservative schemes, thanks to conservative time integrations and consistent numerical fluxes in the system. Moreover, detonation is an extreme singular mode of flame propagation in premixed gas, and the model contains a significant stiff source. It is well known that for hyperbolic equations with stiff source, the transition points in the numerical approximations near the shocks may trigger spurious shock speed, leading to wrong shock position. Intuitively, the high-order weighted essentially non-oscillatory (WENO) scheme, which can suppress oscillations near the discontinuities, would be a good choice for spatial discretization. However, with the nonlinear weights, the numerical fluxes are no longer “consistent”, leading to nonconservative numerical schemes and the bound-preserving technique does not work. Numerical experiments demonstrate that, without further numerical techniques such as subcell resolutions, the conservative FD method with linear weights can yield better numerical approximations than the nonconservative WENO scheme.
In this paper, a new type of finite difference mapped weighted essentially non-oscillatory (MWENO) schemes with unequal-sized stencils, such as the seventh-order and ninth-order versions, is constructed for solving hyperbolic conservation laws. For the purpose of designing increasingly high-order finite difference WENO schemes, the equal-sized stencils are becoming more and more wider. The more we use wider candidate stencils, the bigger the probability of discontinuities lies in all stencils. Therefore, one innovation of these new WENO schemes is to introduce a new splitting stencil methodology to divide some four-point or five-point stencils into several smaller three-point stencils. By the usage of this new methodology in high-order spatial reconstruction procedure, we get different degree polynomials defined on these unequal-sized stencils, and calculate the linear weights, smoothness indicators, and nonlinear weights as specified in Jiang and Shu (J. Comput. Phys. 126: 202228, 1996). Since the difference between the nonlinear weights and the linear weights is too big to keep the optimal order of accuracy in smooth regions, another crucial innovation is to present the new mapping functions which are used to obtain the mapped nonlinear weights and decrease the difference quantity between the mapped nonlinear weights and the linear weights, so as to keep the optimal order of accuracy in smooth regions. These new MWENO schemes can also be applied to compute some extreme examples, such as the double rarefaction wave problem, the Sedov blast wave problem, and the Leblanc problem with a normal CFL number. Extensive numerical results are provided to illustrate the good performance of the new finite difference MWENO schemes.
We develop and use a novel mixed-precision weighted essentially non-oscillatory (WENO) method for solving the Teukolsky equation, which arises when modeling perturbations of Kerr black holes. We show that WENO methods outperform higher-order finite-difference methods, standard in the discretization of the Teukolsky equation, due to the need to add dissipation for stability purposes in the latter. In particular, as the WENO scheme uses no additional dissipation, it is well suited for scenarios requiring long-time evolution such as the study of price tails and gravitational wave emission from extreme mass ratio binaries. In the mixed-precision approach, the expensive computation of the WENO weights is performed in reduced floating-point precision that results in a significant speedup factor of $\approx 3.3$. In addition, we use state-of-the-art Nvidia general-purpose graphics processing units and cluster parallelism to further accelerate the WENO computations. Our optimized WENO solver can be used to quickly generate accurate results of significance in the field of black hole and gravitational wave physics. We apply our solver to study the behavior of the Aretakis charge—a conserved quantity, that if detected by a gravitational wave observatory like LIGO/Virgo would prove the existence of extremal black holes.
In this paper, we present a semi-Lagrangian (SL) method based on a non-polynomial function space for solving the Vlasov equation. We find that a non-polynomial function based scheme is suitable to the specifics of the target problems. To address issues that arise in phase space models of plasma problems, we develop a weighted essentially non-oscillatory (WENO) scheme using trigonometric polynomials. In particular, the non-polynomial WENO method is able to achieve improved accuracy near sharp gradients or discontinuities. Moreover, to obtain a high-order of accuracy in not only space but also time, it is proposed to apply a high-order splitting scheme in time. We aim to introduce the entire SL algorithm with high-order splitting in time and high-order WENO reconstruction in space to solve the Vlasov-Poisson system. Some numerical experiments are presented to demonstrate robustness of the proposed method in having a high-order of convergence and in capturing non-smooth solutions. A key observation is that the method can capture phase structure that require twice the resolution with a polynomial based method. In 6D, this would represent a significant savings.
We address the issue of point value reconstructions from cell averages in the context of third-order finite volume schemes, focusing in particular on the cells close to the boundaries of the domain. In fact, most techniques in the literature rely on the creation of ghost cells outside the boundary and on some form of extrapolation from the inside that, taking into account the boundary conditions, fills the ghost cells with appropriate values, so that a standard reconstruction can be applied also in the boundary cells. In Naumann et al. (Appl. Math. Comput. 325: 252–270. https://doi.org/10.1016/j.amc.2017.12.041, 2018), motivated by the difficulty of choosing appropriate boundary conditions at the internal nodes of a network, a different technique was explored that avoids the use of ghost cells, but instead employs for the boundary cells a different stencil, biased towards the interior of the domain. In this paper, extending that approach, which does not make use of ghost cells, we propose a more accurate reconstruction for the one-dimensional case and a two-dimensional one for Cartesian grids. In several numerical tests, we compare the novel reconstruction with the standard approach using ghost cells.
In this paper, a new kind of hybrid method based on the weighted essentially non-oscillatory (WENO) type reconstruction is proposed to solve hyperbolic conservation laws. Comparing the WENO schemes with/without hybridization, the hybrid one can resolve more details in the region containing multi-scale structures and achieve higher resolution in the smooth region; meanwhile, the essentially oscillation-free solution could also be obtained. By adapting the original smoothness indicator in the WENO reconstruction, the stencil is distinguished into three types: smooth, non-smooth, and high-frequency region. In the smooth region, the linear reconstruction is used and the non-smooth region with the WENO reconstruction. In the high-frequency region, the mixed scheme of the linear and WENO schemes is adopted with the smoothness amplification factor, which could capture high-frequency wave efficiently. Spectral analysis and numerous examples are presented to demonstrate the robustness and performance of the hybrid scheme for hyperbolic conservation laws.
Several important PDE systems, like magnetohydrodynamics and computational electrodynamics, are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion. Recently, new classes of PDE systems have emerged for hyperelasticity, compressible multiphase flows, so-called first-order reductions of the Einstein field equations, or a novel first-order hyperbolic reformulation of Schrödinger’s equation, to name a few, where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector field. We study the problem of curl constraint-preserving reconstruction as it pertains to the design of mimetic finite volume (FV) WENO-like schemes for PDEs that support a curl-preserving involution. (Some insights into discontinuous Galerkin (DG) schemes are also drawn, though that is not the prime focus of this paper.) This is done for two- and three-dimensional structured mesh problems where we deliver closed form expressions for the reconstruction. The importance of multidimensional Riemann solvers in facilitating the design of such schemes is also documented. In two dimensions, a von Neumann analysis of structure-preserving WENO-like schemes that mimetically satisfy the curl constraints, is also presented. It shows the tremendous value of higher order WENO-like schemes in minimizing dissipation and dispersion for this class of problems. Numerical results are also presented to show that the edge-centered curl-preserving (ECCP) schemes meet their design accuracy. This paper is the first paper that invents non-linearly hybridized curl-preserving reconstruction and integrates it with higher order Godunov philosophy. By its very design, this paper is, therefore, intended to be forward-looking and to set the stage for future work on curl involution-constrained PDEs.
We construct new fifth-order alternative WENO (A-WENO) schemes for the Euler equations of gas dynamics. The new scheme is based on a new adaptive diffusion central-upwind Rankine-Hugoniot (CURH) numerical flux. The CURH numerical fluxes have been recently proposed in [Garg et al. J Comput Phys 428, 2021] in the context of second-order semi-discrete finite-volume methods. The proposed adaptive diffusion CURH flux contains a smaller amount of numerical dissipation compared with the adaptive diffusion central numerical flux, which was also developed with the help of the discrete Rankine-Hugoniot conditions and used in the fifth-order A-WENO scheme recently introduced in [Wang et al. SIAM J Sci Comput 42, 2020]. As in that work, we here use the fifth-order characteristic-wise WENO-Z interpolations to evaluate the fifth-order point values required by the numerical fluxes. The resulting one- and two-dimensional schemes are tested on a number of numerical examples, which clearly demonstrate that the new schemes outperform the existing fifth-order A-WENO schemes without compromising the robustness.
In Li and Ren (Int. J. Numer. Methods Fluids 70: 742–763, 2012), a high-order k-exact WENO finite volume scheme based on secondary reconstructions was proposed to solve the two-dimensional time-dependent Euler equations in a polygonal domain, in which the high-order numerical accuracy and the oscillations-free property can be achieved. In this paper, the method is extended to solve steady state problems imposed in a curved physical domain. The numerical framework consists of a Newton type finite volume method to linearize the nonlinear governing equations, and a geometrical multigrid method to solve the derived linear system. To achieve high-order non-oscillatory numerical solutions, the classical k-exact reconstruction with $k=3$ and the efficient secondary reconstructions are used to perform the WENO reconstruction for the conservative variables. The non-uniform rational B-splines (NURBS) curve is used to provide an exact or a high-order representation of the curved wall boundary. Furthermore, an enlarged reconstruction patch is constructed for every element of mesh to significantly improve the convergence to steady state. A variety of numerical examples are presented to show the effectiveness and robustness of the proposed method.