Multidimensional smoothness indicators for first-order Hamilton-Jacobi equations

Falcone, Maurizio; Paolucci, Giulio; Tozza, Silvia

doi:10.1016/j.jcp.2020.109360

Cited by 4 publications

(1 citation statement)

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“…[15,17,18], and many of these ideas extend to HJ equations. We also mention that more recently a new class of high-order filtered schemes has been proposed [5] and improved [12], these schemes converge to the viscosity solution and a precise error estimate has been proved. It can be interesting to deal with discontinuous viscosity solutions so these schemes have to be adapted in order to obtain reasonable approximations which do not diffuse too much around the discontinuities of Du and/or u and do not introduce spurious oscillations.…”

Section: Introductionmentioning

confidence: 96%

Coupled Scheme for Linear and Hamilton-Jacobi Equations: Theoretical and Numerical Aspects

Sahu

2023

Preprint

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We present a comprehensive analysis of the coupled scheme introduced in [Springer Proceedings in Mathematics \& Statistics, vol 237. Springer, Cham 2018 \cite{S2018}] for linear and Hamilton-Jacobi equations. This method merges two distinct schemes, each tailored to handle specific solution characteristics. It offers a versatile framework for coupling various schemes, enabling the integration of accurate methods for smooth solutions and the treatment of discontinuities and gradient jumps. In \cite{S2018}, the emphasis was on coupling an anti-dissipative scheme designed for discontinuous solutions with a semi-Lagrangian scheme developed for smooth solutions. In this paper, we rigorously establish the essential properties of the resulting coupled scheme, especially in the linear case. To illustrate the effectiveness of this coupled approach, we present a series of one-dimensional examples.

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Section: Introductionmentioning

confidence: 96%

Coupled Scheme for Linear and Hamilton-Jacobi Equations: Theoretical and Numerical Aspects

Sahu

2023

Preprint

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Mathematical and Computational Approaches for Stochastic Control of River Environment and Ecology: From Fisheries Viewpoint

Yoshioka

2021

Computational Management

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We present a modern stochastic control framework for dynamic optimization of river environment and ecology. We focus on a fisheries problem in Japan, and show several examples of simplified optimal control problems of stochastic differential equations modeling fishery resource dynamics, reservoir water balance dynamics, benthic algae dynamics, and sediment storage dynamics. These problems concern different phenomena with each other, but they all reduce to solving degenerate parabolic or elliptic equations. Optimal controls and value functions of these problems are computed using finite difference schemes. Finally, we present a higher-dimensional problem of controlling a dam-reservoir system using a semi-Lagrangian discretization on sparse grids. Our contribution shows the state-of-art of modeling, analysis, and computation of stochastic control in environmental engineering and science, and related research areas.

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Efficient Implementation of Adaptive Order Reconstructions

Semplice

Visconti

2020

J Sci Comput

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Including 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 [4,2]. 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.Keywords. CWENOZ-AO -polynomial reconstruction -weighted essentially nonoscillatory -CWENOZadaptive order WENO -finite volume schemes -hyperbolic systemsconservation and balance laws Nq q=0 w q s(R i (t, x q )), where x q and w q are the nodes and weights of a quadrature formula on Ω i .The reconstruction operator should yield an accurate but non oscillatory pointwise approximation of the unknown, based on its cell averages. Beyond the second order of accuracy, one considers essentially non oscillatory reconstructions, which are typically realized by selecting (as in ENO [19]) or more commonly by blending in a nonlinear way polynomials with different degrees and/or stencils. In particular WENO, which was introduced in [33], considers a set of polynomials with equal degree but different stencils and aims at reproducing the accuracy of a higher degree interpolant when the data are locally smooth. The literature on this subject is vast and the reader may refer to [32] for a review.The suboptimal accuracy close to critical points shown by the original design has been later overcome by new definitions of the nonlinear weights (e.g. mapped WENO [20], WENOZ [5,9], the global average weight of [3]) or by taking the small parameter to be dependent on the local mesh size [1].Another source of difficulty in WENO reconstructions is the possible non-existence or nonpositivity of the linear weights for certain grid types and reconstruction points [31]. A quite successful proposal to address this issue was put forward by Levy, Puppo and Russo in [26] for the case of achieving a third order accurate reconstruction at cell center in one and two-dimensional uniform grids. Their CWENO3 reconstruction is a nonlinear blend of one second degree polynomial and of some first degree ones; this approach frees the linear weights from having to satisfy accuracy requirements and allows to choose them arbitrarily, independently of the reconstruction point, independently of the grid type (Ca...

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Multidimensional smoothness indicators for first-order Hamilton-Jacobi equations

Cited by 4 publications

References 31 publications

Coupled Scheme for Linear and Hamilton-Jacobi Equations: Theoretical and Numerical Aspects

Coupled Scheme for Linear and Hamilton-Jacobi Equations: Theoretical and Numerical Aspects

Mathematical and Computational Approaches for Stochastic Control of River Environment and Ecology: From Fisheries Viewpoint

Efficient Implementation of Adaptive Order Reconstructions

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