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 techniq...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.展开更多
Many interesting applications of hyperbolic systems of equations are stiff,and require the time step to satisfy restrictive stability conditions.One way to avoid small time steps is to use implicit time integration.Im...Many interesting applications of hyperbolic systems of equations are stiff,and require the time step to satisfy restrictive stability conditions.One way to avoid small time steps is to use implicit time integration.Implicit integration is quite straightforward for first-order schemes.High order schemes instead also need to control spurious oscillations,which requires limiting in space and time also in the linear case.We propose a framework to simplify considerably the application of high order non-oscillatory schemes through the introduction of a low order implicit predictor,which is used both to set up the nonlinear weights of a standard high order space reconstruction,and to achieve limiting in time.In this preliminary work,we concentrate on the case of a third-order scheme,based on diagonally implicit Runge Kutta(DIRK)integration in time and central weighted essentially non-oscillatory(CWENO)reconstruction in space.The numerical tests involve linear and nonlinear scalar conservation laws.展开更多
基金MIUR-PRIN project 2017KKJP4X“Innovative numerical methods for evolutionary partial differential equations and applications”.Gabriella Puppo acknowledges also the support of 2019 Ateneo Sapienza research project no.RM11916B51CD40E1.
文摘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.
基金MIUR(Ministry of University and Research)PRIN2017 project number 2017KKJP4XProgetto di Ateneo Sapienza,number RM120172B41DBF3A.
文摘Many interesting applications of hyperbolic systems of equations are stiff,and require the time step to satisfy restrictive stability conditions.One way to avoid small time steps is to use implicit time integration.Implicit integration is quite straightforward for first-order schemes.High order schemes instead also need to control spurious oscillations,which requires limiting in space and time also in the linear case.We propose a framework to simplify considerably the application of high order non-oscillatory schemes through the introduction of a low order implicit predictor,which is used both to set up the nonlinear weights of a standard high order space reconstruction,and to achieve limiting in time.In this preliminary work,we concentrate on the case of a third-order scheme,based on diagonally implicit Runge Kutta(DIRK)integration in time and central weighted essentially non-oscillatory(CWENO)reconstruction in space.The numerical tests involve linear and nonlinear scalar conservation laws.
