On new strategies to control the accuracy of WENO algorithms close to discontinuities

This paper is devoted to the construction and analysis of new nonlinear optimal weights for weighted ENO (WENO) interpolation capable of raising the order of accuracy close to discontinuities. The new nonlinear optimal weights are constructed using a strategy inspired by the original WENO algorithm, and they work very well for corner or jump singularities, leading to optimal theoretical accuracy. This is the first part of a series of two papers. In this first part we analyze the performance of the new algorithms proposed for univariate function approximation in the point values (interpolation problem). In the second part, we will extend the analysis to univariate function approximation in the cell averages (reconstruction problem). Our aim is twofold: to raise the order of accuracy of the WENO type interpolation schemes both near discontinuities and in the interval which contains the singularity. The first problem can be solved using the new nonlinear optimal weights, but the second one requires a new strategy that locates the position of the singularity inside the cell in order to attain adaption. This new strategy is inspired by the ENO-SR schemes proposed by Harten [J. Comput. Phys., 83 (1989), pp. 148--184]. Thus, we will introduce two different algorithms in the point values. The first one can deal with corner singularities and jump discontinuities for intervals not containing the singularity. The second algorithm can also deal with intervals containing corner singularities, as they can be detected from the point values, but jump discontinuities cannot, as the information of their position is lost during the discretization process. As mentioned before, the second part of this work will be devoted to the cell averages and, in this context, it will be possible to work with jump discontinuities as well.The work of the authors was supported by the Programa de Apoyo a la Investigatión de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 20928/PI/18, by the national research project MTM2015-64382-P (MINECO/FEDER), and by National Science Foundation grant DMS-1719410

High-order conservative finite difference GLM-MHD schemes for cell-centered MHD

We present and compare third- as well as fifth-order accurate finite difference schemes for the numerical solution of the compressible ideal MHD equations in multiple spatial dimensions. The selected methods lean on four different reconstruction techniques based on recently improved versions of the weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving (MP) schemes as well as slope-limited polynomial reconstruction. The proposed numerical methods are highly accurate in smooth regions of the flow, avoid loss of accuracy in proximity of smooth extrema and provide sharp non-oscillatory transitions at discontinuities. We suggest a numerical formulation based on a cell-centered approach where all of the primary flow variables are discretized at the zone center. The divergence-free condition is enforced by augmenting the MHD equations with a generalized Lagrange multiplier yielding a mixed hyperbolic/parabolic correction, as in Dedner et al. (J. Comput. Phys. 175 (2002) 645-673). The resulting family of schemes is robust, cost-effective and straightforward to implement. Compared to previous existing approaches, it completely avoids the CPU intensive workload associated with an elliptic divergence cleaning step and the additional complexities required by staggered mesh algorithms. Extensive numerical testing demonstrate the robustness and reliability of the proposed framework for computations involving both smooth and discontinuous features.Comment: 32 pages, 14 figure, submitted to Journal of Computational Physics (Aug 7 2009

Adaptive Mesh Refinement for Hyperbolic Systems based on Third-Order Compact WENO Reconstruction

In this paper we generalize to non-uniform grids of quad-tree type the Compact WENO reconstruction of Levy, Puppo and Russo (SIAM J. Sci. Comput., 2001), thus obtaining a truly two-dimensional non-oscillatory third order reconstruction with a very compact stencil and that does not involve mesh-dependent coefficients. This latter characteristic is quite valuable for its use in h-adaptive numerical schemes, since in such schemes the coefficients that depend on the disposition and sizes of the neighboring cells (and that are present in many existing WENO-like reconstructions) would need to be recomputed after every mesh adaption. In the second part of the paper we propose a third order h-adaptive scheme with the above-mentioned reconstruction, an explicit third order TVD Runge-Kutta scheme and the entropy production error indicator proposed by Puppo and Semplice (Commun. Comput. Phys., 2011). After devising some heuristics on the choice of the parameters controlling the mesh adaption, we demonstrate with many numerical tests that the scheme can compute numerical solution whose error decays as $\langle N\rangle^{-3}$, where $\langle N\rangle$ is the average number of cells used during the computation, even in the presence of shock waves, by making a very effective use of h-adaptivity and the proposed third order reconstruction.Comment: many updates to text and figure

Cell-average WENO with progressive order of accuracy close to discontinuities with applications to signal processing

In this paper we translate to the cell-average setting the algorithm for the point-value discretization presented in Amat el al. (2020). This new strategy tries to improve the results of WENO-(2r−1) algorithm close to the singularities, resulting in an optimal order of accuracy at these zones. The main idea is to modify the optimal weights so that they have a nonlinear expression that depends on the position of the discontinuities. In this paper we study the application of the new algorithm to signal processing using Harten’s multiresolution. Several numerical experiments are performed in order to confirm the theoretical results obtained.This work was funded by project 20928/PI/18 (Proyecto financiado por la Comunidad Autónoma de la Región de Murcia a través de la convocatoria de Ayudas a proyectos para el desarrollo de investigación científica y técnica por grupos competitivos, incluida en el Programa Regional de Fomento de la Investigación Científica y Técnica (Plan de Actuación 2018) de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia) and through the national research project (MINECO/FEDER) PID2019-108336GB-I00. 2 The author has been supported through the NSF grant DMS-2010107 and AFOSR grant FA9550-20-1-0055. 3 The author has been supported through the Spanish MINECO project MTM2017-83942-P

Compact-Reconstruction Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws

A new class of non-linear compact interpolation schemes is introduced in this dissertation that have a high spectral resolution and are non-oscillatory across discontinuities. The Compact-Reconstruction Weighted Essentially Non-Oscillatory (CRWENO) schemes use a solution-dependent combination of lower-order compact schemes to yield a high-order accurate, non-oscillatory scheme. Fifth-order accurate CRWENO schemes are constructed and their numerical properties are analyzed. These schemes have lower absolute errors and higher spectral resolution than the WENO scheme of the same order. The schemes are applied to scalar conservation laws and the Euler equations of fluid dynamics. The order of convergence and the higher accuracy of the CRWENO schemes are verified for smooth solutions. Significant improvements are observed in the resolution of discontinuities and extrema as well as the preservation of flow features over large convection distances. The computational cost of the CRWENO schemes is assessed and the reduced error in the solution outweighs the additional expense of the implicit scheme, thus resulting in higher numerical efficiency. This conclusion extends to the reconstruction of conserved and primitive variables for the Euler equations, but not to the characteristic-based reconstruction. Further improvements are observed in the accuracy and resolution of the schemes with alternative formulations for the non-linear weights. The CRWENO schemes are integrated into a structured, finite-volume Navier-Stokes solver and applied to problems of practical relevance. Steady and unsteady flows around airfoils are solved to validate the scheme for curvi-linear grids, as well as overset grids with relative motion. The steady flow around a three-dimensional wing and the unsteady flow around a full-scale rotor are solved. It is observed that though lower-order schemes suffice for the accurate prediction of aerodynamic forces, the CRWENO scheme yields improved resolution of near-blade and wake flow features, including boundary and shear layers, and shed vortices. The high spectral resolution, coupled with the non-oscillatory behavior, indicate their suitability for the direct numerical simulation of compressible turbulent flows. Canonical flow problems -- the decay of isotropic turbulence and the shock-turbulence interaction -- are solved. The CRWENO schemes show an improved resolution of the higher wavenumbers and the small-length-scale flow features that are characteristic of turbulent flows. Overall, the CRWENO schemes show significant improvements in resolving and preserving flow features over a large range of length scales due to the higher spectral resolution and lower dissipation and dispersion errors, compared to the WENO schemes. Thus, these schemes are a viable alternative for the numerical simulation of compressible, turbulent flows

Efficient Finite Difference WENO Scheme for Hyperbolic Systems with Non-Conservative Products

Higher order finite difference Weighted Essentially Non-Oscillatory (WENO) schemes have been constructed for conservation laws. For multidimensional problems, they offer high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of comparable accuracy. This makes them quite attractive for several science and engineering applications. But, to the best of our knowledge, such schemes have not been extended to non-linear hyperbolic systems with non-conservative products. In this paper, we perform such an extension which improves the domain of applicability of such schemes. The extension is carried out by writing the scheme in fluctuation form. We use the HLLI Riemann solver of Dumbser and Balsara (2016) as a building block for carrying out this extension. Because of the use of an HLL building block, the resulting scheme has a proper supersonic limit. The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme, thus expanding its domain of applicability. Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions, making it very easy for users to transition over to the present formulation. For conservation laws, the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO, with two major advantages:- 1) It can capture jumps in stationary linearly degenerate wave families exactly. 2) It only requires the reconstruction to be applied once. Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of accuracy for smooth multidimensional flows. Stringent Riemann ... *Abstract truncated, see PDF*Comment: Accepted in Communications on Applied Mathematics and Computatio