1,721,017 research outputs found
Semi-conservative high order scheme with numerical entropy indicator for intrusive formulations of hyperbolic systems
No full textThis work considers high order discretizations for the intrusive stochastic Galerkin and polynomial moment method. Applications to hyperbolic systems result in solutions that typically involve a large number of wave interactions that must be resolved numerically. In order to reduce numerical oscillations, analytical and numerical entropy indicators are used to perform CWENO-type reconstructions in characteristic variables, when and where non-smooth solutions arise. The proposed method is analyzed for random isentropic Euler equations. In particular, a semi-conservative scheme is employed for non-polynomial pressure in order to reduce the computational cost, while still ensuring correct shock speeds
Surface Reconstruction From Point Cloud using a Semi-Lagrangian Scheme with Local Interpolator
No full textWe propose a level set method to reconstruct unknown surfaces from point clouds, without assuming that the connections between points are known. We consider a variational formulation with a curvature constraint that minimizes the surface area weighted by the distance of the surface from the point cloud. More precisely we solve an equivalent advection–diffusion equation that governs the evolution of an initial surface described implicitly by a level set function. Among all the possible representations, we aim to compute the signed distance function at least in the vicinity of the reconstructed surface. The numerical method for the approximation of the solution is based on a semi-Lagrangian scheme coupled with a local interpolator. In particular, we resort to a multi-linear interpolator and to a Weighted Essentially Non-oscillatory one, to improve the accuracy of the reconstruction. An analysis of the parameters employed in the model is given, focusing in particular on the effect of the curvature regularization, and on the presence of noisy data. Special attention has been paid to the localization of the method and to the development of fast algorithms that run in parallel, resulting in faster reconstruction and thus the opportunity to easily improve the resolution. Numerical tests in two and three dimensions are presented to evaluate the quality of the reconstruction and the efficiency of the algorithm in terms of computational time
Well balanced high order 1D schemes on non-uniform grids and entropy residuals
Full text linkThis paper is concerned with the construction of high order schemes on irregular grids for balance laws, including a discussion of an a-posteriori error indicator based on the numerical entropy production. We also impose well-balancing on non uniform grids for the shallow water equations, which can be extended similarly to other balance laws, obtaining schemes up to fourth order of accuracy with very weak assumptions on the regularity of the grid. Our results show the expected convergence rates, the correct propagation of shocks across grid discontinuities and demonstrate the improved resolution achieved with a locally refined non-uniform grid. The schemes proposed in this work naturally can also be applied to systems of conservation laws. They may also be extended to higher space dimensions by means of dimensional splitting. The error indicator based on the numerical entropy production, previously introduced for the case of systems of conservation laws, is extended to balance laws. Its decay rate and its ability to identify discontinuities is illustrated on several tests
One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells
Full text linkWe 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.org110.1016/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
Adaptive mesh refinement for hyperbolic systems based on third-order Compact WENO reconstruction
Full text linkIn this paper we generalise to non-uniform grids of quad-tree type the Compact WENO reconstruction of Levy et al. (SIAM J Sci Comput 22(2):656–672, 2000), 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 neighbouring 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 10(5):1132–1160, 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 ⟨N⟩−3⟨N⟩−3, where ⟨N⟩⟨N⟩ 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
Arbitrary order finite volume well-balanced schemes for the Euler equations with gravity
Full text linkThis work presents arbitrary high order well balanced finite volume schemes for the Euler equations with a prescribed gravitational field. It is assumed that the desired equilibrium solution is known, and we construct a scheme which is exactly well balanced for that particular equilibrium. The scheme is based on high order reconstructions of the fluctuations from equilibrium of density, velocity, and pressure, and on a well-balanced integration of the source terms, while no assumptions are needed on the numerical flux, beside consistency. This technique also allows one to construct well-balanced methods for a class of moving equilibria. Several numerical tests demonstrate the performance of the scheme on different scenarios, from equilibrium solutions to nonsteady problems involving shocks. The numerical tests are carried out with methods up to fifth order in one dimension, and third order accuracy in two dimensions
AMG preconditioning for nonlinear degenerate parabolic equations on nonuniform grids with application to monument degradation
No full textMotivated by the modelling of marble degradation by chemical pollutants, we consider the ap- proximation by implicit finite differences schemes of nonlinear degenerate parabolic equations in which sharp boundary layers naturally occur. The latter suggests to consider various types of nonuniform griddings, when defining suitable approximation schemes. The resulting large nonlinear systems are treated by Newton methods, while the locally Toeplitz linear systems aris- ing from the Jacobian have to be solved efficiently. To this end, we propose the use of AMG preconditioners and we study the related convergence issues, together with the associated spec- tral features. We present some numerical experiments supporting our theoretical results on the spectrum of the coefficient matrix of the linear systems, alongside others regarding the numerical simulations in the case of the specific model
Quinpi: Integrating Stiff Hyperbolic Systems with Implicit High Order Finite Volume Schemes
No full textMany interesting physical problems described by systems of hyperbolic conservation laws are stiff, and thus impose a very small time-step because of the restrictive CFL stability condition. In this case, one can exploit the superior stability properties of implicit time integration which allows to choose the time-step only from accuracy requirements, and thus avoid the use of small time-steps. We discuss an efficient framework to devise high order implicit schemes for stiff hyperbolic systems without tailoring it to a specific problem. The nonlinearity of high order schemes, due to space- and time-limiting procedures which control nonphysical oscillations, makes the implicit time integration difficult, e.g. because the discrete system is nonlinear also on linear problems. This nonlinearity of the scheme is circumvented as proposed in (Puppo et al., Comm. Appl. Math. & Comput., 2023) for scalar conservation laws, where a first order implicit predictor is computed to freeze the nonlinear coefficients of the essentially non-oscillatory space reconstruction, and also to assist limiting in time. In addition, we propose a novel conservative flux-centered a-posteriori time-limiting procedure using numerical entropy indicators to detect troubled cells. The numerical tests involve classical and artificially devised stiff problems using the Euler's system of gas-dynamics
Efficient Implementation of Adaptive Order Reconstructions
Full text linkIncluding polynomials with small degree and stencil when designing very high order reconstructions is surely beneficial for their non oscillatory properties, but may bring loss of accuracy on smooth data unless special care is exerted. In this paper we address this issue with a new Central WENOZ (CWENOZ) approach, in which the reconstruction polynomial is computed from a single set of non linear weights, but the linear weights of the polynomials with very low degree (compared to the final desired accuracy) are infinitesimal with respect to the grid size. After proving general results that guide the choice of the CWENOZ parameters, we study a concrete example of a reconstruction that blends polynomials of degree six, four and two, mimicking already published Adaptive Order WENO reconstructions (Arbogast et al. in SIAM J Numer Anal 56(3):1818-1947, 2018),(Balsara et al. in J Comput Phys 326:780-804, 2016). The novel reconstruction yields similar accuracy and oscillations with respect to the previous ones, but saves up to 20% computational time since it does not rely on a hierarchic approach and thus does not compute multiple sets of nonlinear weights in each cell
Analysis of multigrid preconditioning for implicit PDE solvers for degenerate parabolic equations
No full textIn this paper an implicit numerical method designed for nonlinear degenerate parabolic equations is proposed. A convergence analysis and the study of the related computational cost are provided. In fact, due to the nonlinear nature of the underlying mathematical model, the use of a fixed point scheme is required. The chosen scheme is the Newton method and its convergence is proven under mild assumptions. Every step of the Newton method implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate multigrid preconditioned Krylov methods. Numerical experiments for the validation of our analysis complement this contribution
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