In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial deriv...In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial derivative term and the forward and backward Euler method to discretize the time derivative term, the explicit and implicit upwind difference schemes are obtained respectively. It is proved that the explicit upwind scheme is conditionally stable and the implicit upwind scheme is unconditionally stable. Then the convergence of the schemes is derived. Numerical examples verify the results of theoretical analysis.展开更多
A posteriori error estimate of the discontinuous-streamline diffusion method for first-order hyperbolic equations was presented, which can be used to adjust space mesh reasonably. A numerical example is given to illus...A posteriori error estimate of the discontinuous-streamline diffusion method for first-order hyperbolic equations was presented, which can be used to adjust space mesh reasonably. A numerical example is given to illustrate the accuracy and feasibility of this method.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
A new modification of the Method of Lines is proposed for the solution of first order partial differential equations. The accuracy of the method is shown with the matrix analysis. The method is applied to a number of ...A new modification of the Method of Lines is proposed for the solution of first order partial differential equations. The accuracy of the method is shown with the matrix analysis. The method is applied to a number of test problems, on uniform grids, to compare the accuracy and computational efficiency with the standard method.展开更多
Models for shallow water flow often assume that the lateral velocity is constant over the water height.The recently derived shallow water moment equations are an extension of these standard shallow water equations.The...Models for shallow water flow often assume that the lateral velocity is constant over the water height.The recently derived shallow water moment equations are an extension of these standard shallow water equations.The extended models allow for a vertically changing velocity profile,resulting in more accuracy when the velocity varies considerably over the height of the fluid.Unfortunately,already the one-dimensional models lack global hyperbolicity,an important property of partial differential equations that ensures that disturbances have a finite propagation speed.In this paper,cylindrical shallow water moment equations are formulated by starting from the cylindrical incompressible Navier-Stokes equations.We formulate twodimensional axisymmetric ShallowWater Moment Equations by imposing axisymmetry in the cylindrical model.The loss of hyperbolicity is analyzed and a hyperbolic axisymmetric moment model is then derived by modifying the system matrix in analogy to the one-dimensional case,for which the hyperbolicity problem has already been observed and overcome.Numerical simulations with both discontinuous and continuous initial data in a cylindrical domain are performed using a finite volume scheme tailored to the cylindrical mesh.The newly derived hyperbolic model is clearly beneficial as it gives more stable solutions and still converges to the reference solution when increasing the number of moments.展开更多
In this paper,we shall carry out the L^(2)-norm stability analysis of the Runge-Kutta discontinuous Galerkin(RKDG)methods on rectangle meshes when solving a linear constant-coefficient hyperbolic equation.The matrix t...In this paper,we shall carry out the L^(2)-norm stability analysis of the Runge-Kutta discontinuous Galerkin(RKDG)methods on rectangle meshes when solving a linear constant-coefficient hyperbolic equation.The matrix transferring process based on temporal differences of stage solutions still plays an important role to achieve a nice energy equation for carrying out the energy analysis.This extension looks easy for most cases;however,there are a few troubles with obtaining good stability results under a standard CFL condition,especially,for those Q^(k)-elements with lower degree k as stated in the one-dimensional case.To overcome this difficulty,we make full use of the commutative property of the spatial DG derivative operators along two directions and set up a new proof line to accomplish the purpose.In addition,an optimal error estimate on Q^(k)-elements is also presented with a revalidation on the supercloseness property of generalized Gauss-Radau(GGR)projection.展开更多
We consider an inverse problem of determining unknown coefficients for a one-dimensional analogue of radiative transport equation.We show that some combination of the unknown coefficients can be uniquely determined by...We consider an inverse problem of determining unknown coefficients for a one-dimensional analogue of radiative transport equation.We show that some combination of the unknown coefficients can be uniquely determined by giving pulse-like inputs at the boundary and observing the corresponding outputs.Our result can be applied for determination of absorption and scattering properties of an optically turbid medium if the radiative transport equation is appropriate for describing the propagation of light in the medium.展开更多
文摘In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial derivative term and the forward and backward Euler method to discretize the time derivative term, the explicit and implicit upwind difference schemes are obtained respectively. It is proved that the explicit upwind scheme is conditionally stable and the implicit upwind scheme is unconditionally stable. Then the convergence of the schemes is derived. Numerical examples verify the results of theoretical analysis.
文摘A posteriori error estimate of the discontinuous-streamline diffusion method for first-order hyperbolic equations was presented, which can be used to adjust space mesh reasonably. A numerical example is given to illustrate the accuracy and feasibility of this method.
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A new modification of the Method of Lines is proposed for the solution of first order partial differential equations. The accuracy of the method is shown with the matrix analysis. The method is applied to a number of test problems, on uniform grids, to compare the accuracy and computational efficiency with the standard method.
文摘Models for shallow water flow often assume that the lateral velocity is constant over the water height.The recently derived shallow water moment equations are an extension of these standard shallow water equations.The extended models allow for a vertically changing velocity profile,resulting in more accuracy when the velocity varies considerably over the height of the fluid.Unfortunately,already the one-dimensional models lack global hyperbolicity,an important property of partial differential equations that ensures that disturbances have a finite propagation speed.In this paper,cylindrical shallow water moment equations are formulated by starting from the cylindrical incompressible Navier-Stokes equations.We formulate twodimensional axisymmetric ShallowWater Moment Equations by imposing axisymmetry in the cylindrical model.The loss of hyperbolicity is analyzed and a hyperbolic axisymmetric moment model is then derived by modifying the system matrix in analogy to the one-dimensional case,for which the hyperbolicity problem has already been observed and overcome.Numerical simulations with both discontinuous and continuous initial data in a cylindrical domain are performed using a finite volume scheme tailored to the cylindrical mesh.The newly derived hyperbolic model is clearly beneficial as it gives more stable solutions and still converges to the reference solution when increasing the number of moments.
基金supported by the NSFC(Grant No.12301513)the Natural Science Foundation of Jiangsu Province,China(Grant No.BK20230374)+1 种基金the Natural Science Foundation of Jiangsu Higher Education Institutions of China(Grant No.23KJB110019)supported by the NSFC(Grant No.12071214).
文摘In this paper,we shall carry out the L^(2)-norm stability analysis of the Runge-Kutta discontinuous Galerkin(RKDG)methods on rectangle meshes when solving a linear constant-coefficient hyperbolic equation.The matrix transferring process based on temporal differences of stage solutions still plays an important role to achieve a nice energy equation for carrying out the energy analysis.This extension looks easy for most cases;however,there are a few troubles with obtaining good stability results under a standard CFL condition,especially,for those Q^(k)-elements with lower degree k as stated in the one-dimensional case.To overcome this difficulty,we make full use of the commutative property of the spatial DG derivative operators along two directions and set up a new proof line to accomplish the purpose.In addition,an optimal error estimate on Q^(k)-elements is also presented with a revalidation on the supercloseness property of generalized Gauss-Radau(GGR)projection.
文摘We consider an inverse problem of determining unknown coefficients for a one-dimensional analogue of radiative transport equation.We show that some combination of the unknown coefficients can be uniquely determined by giving pulse-like inputs at the boundary and observing the corresponding outputs.Our result can be applied for determination of absorption and scattering properties of an optically turbid medium if the radiative transport equation is appropriate for describing the propagation of light in the medium.
