We extend LeVeque's wave propagation algorithm,a widely used finite volume method for hyperbolic partial differential equations,to a third-order accurate method.The resulting scheme shares main properties with the...We extend LeVeque's wave propagation algorithm,a widely used finite volume method for hyperbolic partial differential equations,to a third-order accurate method.The resulting scheme shares main properties with the original method,i.e.,it is based on a wave decomposition at grid cell interfaces,it can be used to approximate hyperbolic problems in divergence form as well as in quasilinear form and limiting is introduced in the form of a wave limiter.展开更多
The leaderless and leader-following finite-time consensus problems for multiagent systems(MASs)described by first-order linear hyperbolic partial differential equations(PDEs)are studied.The Lyapunov theorem and the un...The leaderless and leader-following finite-time consensus problems for multiagent systems(MASs)described by first-order linear hyperbolic partial differential equations(PDEs)are studied.The Lyapunov theorem and the unique solvability result for the first-order linear hyperbolic PDE are used to obtain some sufficient conditions for ensuring the finite-time consensus of the leaderless and leader-following MASs driven by first-order linear hyperbolic PDEs.Finally,two numerical examples are provided to verify the effectiveness of the proposed methods.展开更多
This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discret...This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discretization-then-continuousization is proposed in this paper to cope with the infinite-dimensional nature of PDE systems.The contributions of this paper consist of the following aspects:(1)The differential Riccati equations and the solvability condition of the LQ optimal control problems are obtained via the discretization-then-continuousization method.(2)A numerical calculation way of the differential Riccati equations and a practical design way of the optimal controller are proposed.Meanwhile,the relationship between the optimal costate and the optimal state is established by solving a set of forward and backward partial difference equations(FBPDEs).(3)The correctness of the method used in this paper is verified by a complementary continuous method and the comparative analysis with the existing operator results is presented.It is shown that the proposed results not only contain the classic results of the standard LQ control problem of systems governed by ordinary differential equations as a special case,but also support the existing operator results and give a more convenient form of computation.展开更多
High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of th...High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of the WENO algorithm,large amount of computational costs are required for solving multidimensional problems.In our previous work(Lu et al.in Pure Appl Math Q 14:57–86,2018;Zhu and Zhang in J Sci Comput 87:44,2021),sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations,and it was shown that significant CPU times were saved,while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids.In this technical note,we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme,which has very interesting properties such as its simplicity in linear weights’construction over a classical WENO scheme.Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times,and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.展开更多
In this paper we develop a new closure theory for moment approximationsin kinetic gas theory and derive hyperbolic moment equations for 13 fluid variablesincluding stress and heat flux. Classical equations have either...In this paper we develop a new closure theory for moment approximationsin kinetic gas theory and derive hyperbolic moment equations for 13 fluid variablesincluding stress and heat flux. Classical equations have either restricted hyperbolicity regions like Grad’s moment equations or fail to include higher moments in apractical way like the entropy maximization approach. The new closure is based onPearson-Type-IV distributions which reduce to Maxwellians in equilibrium, but allowanisotropies and skewness in non-equilibrium. The closure relations are essentiallyexplicit and easy to evaluate. Hyperbolicity is shown numerically for a large range ofvalues. Numerical solutions of Riemann problems demonstrate the capability of thenew equations to handle strong non-equilibrium.展开更多
By using fluid dynamics theory with the effects of adsorption and reaction, the chromatography model with a reaction A →B was established as a system of two hyperbolic partial differential equations (PDE’s)....By using fluid dynamics theory with the effects of adsorption and reaction, the chromatography model with a reaction A →B was established as a system of two hyperbolic partial differential equations (PDE’s). In some practical situations, the reaction chromatography model was simplified a semi-coupled system of two linear hyperbolic PDE’s. In which, the reactant concentration wave model was the initial-boundary value problem of a self-closed hyperbolic PDE, while the resultant concentration wave model was the initial-boundary value problem of hyperbolic PDE coupling reactant concentration. The general explicit expressions for the concentration wave of the reactants and resultants were derived by Laplace transform. The δ-pulse and wide pulse injections were taken as the examples to discuss detailedly, and then the stability analysis between the resultant solutions of the two modes of pulse injection was further discussed. It was significant for further analysis of chromatography, optimizing chromatographic separation, determining the physical and chemical characters.展开更多
A construction of multiple knot B-spline wavelets has been given in[C.K.Chui and E.Quak,Wavelet on a bounded interval,In:D.Braess and L.L.Schumaker,editors.Numerical methods of approximation theory.Basel:Birkhauser Ve...A construction of multiple knot B-spline wavelets has been given in[C.K.Chui and E.Quak,Wavelet on a bounded interval,In:D.Braess and L.L.Schumaker,editors.Numerical methods of approximation theory.Basel:Birkhauser Verlag;(1992),pp.57–76].In this work,we first modify these wavelets to solve the elliptic(partially)Dirichlet boundary value problems by Galerkin and Petrov Galerkin methods.We generalize this construction to two dimensional case by Tensor product space.In addition,the solution of the system discretized by Galerkin method with modified multiple knot B-spline wavelets is discussed.We also consider a nonlinear partial differential equation for unsteady flows in an open channel called Saint-Venant.Since the solving of this problem by some methods such as finite difference and finite element produce unsuitable approximations specially in the ends of channel,it is solved by multiple knot B-spline wavelet method that yields a very well approximation.Finally,some numerical examples are given to support our theoretical results.展开更多
基金This work was supported by the DFG through HE 4858/4-1
文摘We extend LeVeque's wave propagation algorithm,a widely used finite volume method for hyperbolic partial differential equations,to a third-order accurate method.The resulting scheme shares main properties with the original method,i.e.,it is based on a wave decomposition at grid cell interfaces,it can be used to approximate hyperbolic problems in divergence form as well as in quasilinear form and limiting is introduced in the form of a wave limiter.
基金the National Natural Science Foundation of China(Nos.11671282 and 12171339)。
文摘The leaderless and leader-following finite-time consensus problems for multiagent systems(MASs)described by first-order linear hyperbolic partial differential equations(PDEs)are studied.The Lyapunov theorem and the unique solvability result for the first-order linear hyperbolic PDE are used to obtain some sufficient conditions for ensuring the finite-time consensus of the leaderless and leader-following MASs driven by first-order linear hyperbolic PDEs.Finally,two numerical examples are provided to verify the effectiveness of the proposed methods.
基金supported by the National Natural Science Foundation of China under Grant Nos.61821004 and 62250056the Natural Science Foundation of Shandong Province under Grant Nos.ZR2021ZD14 and ZR2021JQ24+1 种基金Science and Technology Project of Qingdao West Coast New Area under Grant Nos.2019-32,2020-20,2020-1-4,High-level Talent Team Project of Qingdao West Coast New Area under Grant No.RCTDJC-2019-05Key Research and Development Program of Shandong Province under Grant No.2020CXGC01208.
文摘This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discretization-then-continuousization is proposed in this paper to cope with the infinite-dimensional nature of PDE systems.The contributions of this paper consist of the following aspects:(1)The differential Riccati equations and the solvability condition of the LQ optimal control problems are obtained via the discretization-then-continuousization method.(2)A numerical calculation way of the differential Riccati equations and a practical design way of the optimal controller are proposed.Meanwhile,the relationship between the optimal costate and the optimal state is established by solving a set of forward and backward partial difference equations(FBPDEs).(3)The correctness of the method used in this paper is verified by a complementary continuous method and the comparative analysis with the existing operator results is presented.It is shown that the proposed results not only contain the classic results of the standard LQ control problem of systems governed by ordinary differential equations as a special case,but also support the existing operator results and give a more convenient form of computation.
文摘High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of the WENO algorithm,large amount of computational costs are required for solving multidimensional problems.In our previous work(Lu et al.in Pure Appl Math Q 14:57–86,2018;Zhu and Zhang in J Sci Comput 87:44,2021),sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations,and it was shown that significant CPU times were saved,while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids.In this technical note,we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme,which has very interesting properties such as its simplicity in linear weights’construction over a classical WENO scheme.Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times,and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.
文摘In this paper we develop a new closure theory for moment approximationsin kinetic gas theory and derive hyperbolic moment equations for 13 fluid variablesincluding stress and heat flux. Classical equations have either restricted hyperbolicity regions like Grad’s moment equations or fail to include higher moments in apractical way like the entropy maximization approach. The new closure is based onPearson-Type-IV distributions which reduce to Maxwellians in equilibrium, but allowanisotropies and skewness in non-equilibrium. The closure relations are essentiallyexplicit and easy to evaluate. Hyperbolicity is shown numerically for a large range ofvalues. Numerical solutions of Riemann problems demonstrate the capability of thenew equations to handle strong non-equilibrium.
文摘By using fluid dynamics theory with the effects of adsorption and reaction, the chromatography model with a reaction A →B was established as a system of two hyperbolic partial differential equations (PDE’s). In some practical situations, the reaction chromatography model was simplified a semi-coupled system of two linear hyperbolic PDE’s. In which, the reactant concentration wave model was the initial-boundary value problem of a self-closed hyperbolic PDE, while the resultant concentration wave model was the initial-boundary value problem of hyperbolic PDE coupling reactant concentration. The general explicit expressions for the concentration wave of the reactants and resultants were derived by Laplace transform. The δ-pulse and wide pulse injections were taken as the examples to discuss detailedly, and then the stability analysis between the resultant solutions of the two modes of pulse injection was further discussed. It was significant for further analysis of chromatography, optimizing chromatographic separation, determining the physical and chemical characters.
文摘A construction of multiple knot B-spline wavelets has been given in[C.K.Chui and E.Quak,Wavelet on a bounded interval,In:D.Braess and L.L.Schumaker,editors.Numerical methods of approximation theory.Basel:Birkhauser Verlag;(1992),pp.57–76].In this work,we first modify these wavelets to solve the elliptic(partially)Dirichlet boundary value problems by Galerkin and Petrov Galerkin methods.We generalize this construction to two dimensional case by Tensor product space.In addition,the solution of the system discretized by Galerkin method with modified multiple knot B-spline wavelets is discussed.We also consider a nonlinear partial differential equation for unsteady flows in an open channel called Saint-Venant.Since the solving of this problem by some methods such as finite difference and finite element produce unsuitable approximations specially in the ends of channel,it is solved by multiple knot B-spline wavelet method that yields a very well approximation.Finally,some numerical examples are given to support our theoretical results.