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

Simple smoothness indicator WENO-Z scheme for hyperbolic conservation laws

The advantage of WENO-JS5 scheme [ J. Comput. Phys. 1996] over the WENO-LOC scheme [J. Comput. Phys.1994] is that the WENO-LOC nonlinear weights do not achieve the desired order of convergence in smooth monotone regions and at critical points. In this article, this drawback is achieved with the WENO-LOC smoothness indicators by constructing a WENO-Z type nonlinear weights which contains a novel global smoothness indicator. This novel smoothness indicator measures the derivatives of the reconstructed flux in a global stencil, as a result, the proposed numerical scheme could decrease the dissipation near the discontinuous regions. The theoretical and numerical experiments to achieve the required order of convergence in smooth monotone regions, at critical points, the essentially non-oscillatory (ENO), the analysis of parameters involved in the nonlinear weights like $\epsilon$ and $p$ are studied. From this study, we conclude that the imposition of certain conditions on $\epsilon$ and $p$, the proposed scheme achieves the global order of accuracy in the presence of an arbitrary number of critical points. Numerical tests for scalar, one and two-dimensional system of Euler equations are presented to show the effective performance of the proposed numerical scheme.Comment: 25 pages, 10 figure

On the nature of weighted essentially non-oscillatory scheme based on mach number

Orientadores: Sávio Souza Venâncio Vianna, Rogério Gonçalves dos SantosTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia QuímicaResumo: A solução numérica de problemas de escoamento contento elevados gradientes de pressão, ondas de choque e descontinuidade transientes exige métodos de alta resolução e pouco dissipativo. Nesses casos, estratégias dependente de parâmetros, tais como viscosidade artificial ou limitadores de fluxos, não são as melhores opções. Dentre os esquemas de alta resolução, destaca-se o esquema da família WENO (do inglês, Weighted Essencially Non-Oscillatory), sendo no mínimo de terceira ordem, essencialmente não oscilatório e dotado de um indicador de suavidade calculado a partir das características do escoamento. O esquema WENO de diferenças finitas, utilizado nesta tese, baseia-se na reconstrução por sub-estêncil, a partir de uma malha estruturada e uniforme. Neste trabalho, modificou-se o esquema WENO-Z+ e propôs-se um novo esquema WENO baseado no número de Mach, chamado de WENO-SV. A modificação seguiu a proposição geral do trabalho realizado por Acker et al. (2016), segundo a qual as contribuições para os sub-estênceis onde a solução é menos suave é mais relevante. O esquema WENO-SV foi validado com experimentos numéricos benchmark e aplicado em casos de escoamento em bocais. Além disso, foram propostos modelos físicos acoplados com o esquema WENO-SV para estudo de explosões físicas em ambientes abertos e tubulações. O esquema WENO-SV aumentou, efetivamente, o peso dos sub-estênceis menos suaves gerando melhores resultados do que aqueles apresentados pelo esquema WENO-Z+ para problemas benchmark unidimensionais. Em problemas bidimensionais, o esquema WENO-SV gerou resultados menos dissipativos, entretanto, menos simétricos. Os resultados em casos de escoamentos em bocais mostraram-se satisfatórios. O esquema WENO-SV mostrou-se mais robusto que o método tradicional MacCormack e ainda foi possível verificar uma redução significativa do tempo computacional para a simulação de escoamento em bocais ao modificar o método de integração. E, finalmente, os detalhes da onda de choque nas explosões físicas calculadas pelo modelo proposto mostraram-se de acordo com os dados experimentais e analíticos disponíveis Abstract: Numerical solution of flow problems with high pressure gradient, shock waves and transient discontinuity require high resolution and less dissipative methods for faithful representation. For such cases, numerical schemes coupled with artificial viscosity or flow limiters are not the best options. Among the high resolution schemes, WENO family scheme (Weighted Essentially Non-Oscillatory) stands out. The WENO finite difference scheme used in the present research is based on polynomial reconstruction from a uniform and structured mesh. In this work, WENO-Z+ scheme was modified and a new WENO scheme based on the Mach number, named WENO-SV, was proposed. The modification followed a general proposal by Acker et al. (2016) that to increase the weight of the less smooth sub-stencil led to a better WENO scheme resolution. The WENO-SV scheme has been validated with benchmark problems and applied in nozzle flow cases. In addition, physical models coupled with WENO-SV scheme for physical explosion study in open environments and pipelines were proposed. The WENO-SV scheme effectively increased the weight of less smooth sub-stencils more than those presented by the WENO-Z+ scheme for one-dimensional benchmark problems. In two-dimensional problems, the WENO-SV scheme yields less dissipative but less symmetrical results. The results in cases of nozzle flow areDoutoradoEngenharia QuímicaDoutor em Engenharia Química33003017034P8  CAPE

Numerical Methods for Hyperbolic Partial Differential Equations

Department of Mathematical SciencesIn this dissertation, new numerical methods are proposed for different types of hyperbolic partial differential equations (PDEs). The objectives of these developments aim for the improvements in accuracy, robustness, efficiency, and reduction of the computational cost. The dissertation consists of two parts. The first half discusses shock-capturing methods for nonlinear hyperbolic conservation laws, and proposes a new adaptive weighted essentially non-oscillatory WENO-?? scheme in the context of finite difference. Depending on the smoothness of the large stencils used in the reconstruction of the numerical flux, a parameter ?? is set adaptively to switch the scheme between a 5th-order upwind or 6th-order central discretization. A new indicator depending on parameter ?? measures the smoothness of the large stencils in order to choose a smoother one for the reconstruction procedure. ?? is devised based on the possible highest-order variations of the reconstructing polynomials in an L2 sense. In addition, a new set of smoothness indicators ??_k???s of the sub-stencils is introduced. These are constructed in a central sense with respect to the Taylor expansions around point x_j . Numerical results show that the new scheme combines good properties of both 5th-order upwind and 6th-order central schemes. In particular, the new scheme captures discontinuities and resolves small-scaled structures much better than other 5th-order schemes; overcomes the loss of resolution near some critical regions; and is able to maintain symmetry which are drawbacks detected in other 6th-order central WENO schemes. The second part extends the scope to hyperbolic PDEs with uncertainty, and semi-analytical methods using singular perturbation analysis for dispersive PDEs. For the former, a hybrid operator splitting method is developed for computation of the two-dimensional transverse magnetic Maxwell equations in media with multiple random interfaces. By projecting the solutions into random space using the Polynomial Chaos (PC) expansions, the deterministic and random parts of the solution are solved separately. The deterministic parts are then numerically approximated by the FDTD method with domain decomposition implemented on a staggered grid. Statistic quantities are obtained by the Monte Carlo sampling in the post-processing stage. Parallel computing is proposed for which the computational cost grows linearly with the number of random interfaces. The last section deals with spectral methods for dispersive PDEs. The Kortewegde Vries (KdV) equation is chosen as a prototype. By Fourier series, the PDE is transformed into a system of ODEs which is stiff, that is, there are rapid oscillatory modes for large wavenumbers. A new semi-analytical method is proposed to tackle the difficulty. The new method is based on the classical integrating factor (IF) and exponential time differencing (ETD) schemes. The idea is to approximate analytically the stiff parts by the so-called correctors and numerically the non-stiff parts by the IF and ETD methods. It turns out that rapid oscillations are well absorbed by our corrector method, yielding better accuracy in the numerical results. Due to the nonlinearity, all Fourier modes interact with each other, causing the computation of the correctors to be very costly. In order to overcome this, the correctors are recursively constructed to accurately capture the stiffness of the mode interactions.ope