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
A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions. The scheme is based on a fourthorder central weighted essentially nonoscillatory (CW...A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions. The scheme is based on a fourthorder central weighted essentially nonoscillatory (CWENO) reconstruction for one-dimensional cases, which is generalized to two-dimensional cases by the dimension-by-dimension approach. The large stability domain Runge-Kutta-type solver ROCK4 is used for time integration. The resulting method requires neither the use of Riemann solvers nor the computation of Jacobians and therefore it enjoys the main advantage of the relaxation schemes. The high accuracy and high-resolution properties of the present method are demonstrated in one- and two-dimensional numerical experiments.展开更多
In this paper,we present a third-order central weighted essentially nonoscillatory(CWENO)reconstruction for computations of hyperbolic conservation laws in three space dimensions.Simultaneously,as a Godunov-type centr...In this paper,we present a third-order central weighted essentially nonoscillatory(CWENO)reconstruction for computations of hyperbolic conservation laws in three space dimensions.Simultaneously,as a Godunov-type central scheme,the CWENOtype central-upwind scheme,i.e.,the semi-discrete central-upwind scheme based on our third-order CWENO reconstruction,is developed straightforwardly to solve 3D systems by the so-called componentwise and dimensional-by-dimensional technologies.The high resolution,the efficiency and the nonoscillatory property of the scheme can be verified by solving several numerical experiments.展开更多
A numerical two-dimensional shallow water method was based on method for solving the equations was presented. This the third-order genuinely multidimensional semi-discrete central scheme for spatial discretization an...A numerical two-dimensional shallow water method was based on method for solving the equations was presented. This the third-order genuinely multidimensional semi-discrete central scheme for spatial discretization and the optimal third-order Strong Stability Preserving (SSP) Runge-Kutta method for time integration. The third-order compact Central Weighted Essentially Non-Oscillatory (CWENO) reconstruction was adopted to guarantee the non-oscillatory behavior of the presented scheme and improve the resolution. Two kinds of source terms were considered in this work. They were evaluated using different approaches. The resulting scheme does not require Riemann solvers or characteristic decomposition, hence it retains all the attractive features of central schemes such as simplicity and high resolution. To evaluate the performance of the presented scheme, several numerical examples were tested. The results demonstrate that our method is efficient, stable and robust.展开更多
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.
基金the National Natural Science Foundation of China (60134010)The English text was polished by Yunming Chen.
文摘A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions. The scheme is based on a fourthorder central weighted essentially nonoscillatory (CWENO) reconstruction for one-dimensional cases, which is generalized to two-dimensional cases by the dimension-by-dimension approach. The large stability domain Runge-Kutta-type solver ROCK4 is used for time integration. The resulting method requires neither the use of Riemann solvers nor the computation of Jacobians and therefore it enjoys the main advantage of the relaxation schemes. The high accuracy and high-resolution properties of the present method are demonstrated in one- and two-dimensional numerical experiments.
基金supported by the National Natural Science Foundation of China(Grant Nos.11101333,11071196,11171043)the National Natural Science Foundation of Shaanxi(Grant No.2011GQ1018)NPU Foundation for Fundamental Research.
文摘In this paper,we present a third-order central weighted essentially nonoscillatory(CWENO)reconstruction for computations of hyperbolic conservation laws in three space dimensions.Simultaneously,as a Godunov-type central scheme,the CWENOtype central-upwind scheme,i.e.,the semi-discrete central-upwind scheme based on our third-order CWENO reconstruction,is developed straightforwardly to solve 3D systems by the so-called componentwise and dimensional-by-dimensional technologies.The high resolution,the efficiency and the nonoscillatory property of the scheme can be verified by solving several numerical experiments.
基金Project supported by the National Natural Science Foundation of China (Grant No: 60134010).
文摘A numerical two-dimensional shallow water method was based on method for solving the equations was presented. This the third-order genuinely multidimensional semi-discrete central scheme for spatial discretization and the optimal third-order Strong Stability Preserving (SSP) Runge-Kutta method for time integration. The third-order compact Central Weighted Essentially Non-Oscillatory (CWENO) reconstruction was adopted to guarantee the non-oscillatory behavior of the presented scheme and improve the resolution. Two kinds of source terms were considered in this work. They were evaluated using different approaches. The resulting scheme does not require Riemann solvers or characteristic decomposition, hence it retains all the attractive features of central schemes such as simplicity and high resolution. To evaluate the performance of the presented scheme, several numerical examples were tested. The results demonstrate that our method is efficient, stable and robust.