For multidimensional first order semilinear hyperbolic systems of diagonal form without self-interaction,we show the global nonlinear stability of traveling wave solutions.
In this paper,a new efficient,and at the same time,very simple and general class of thermodynamically compatiblefinite volume schemes is introduced for the discretization of nonlinear,overdetermined,and thermodynamicall...In this paper,a new efficient,and at the same time,very simple and general class of thermodynamically compatiblefinite volume schemes is introduced for the discretization of nonlinear,overdetermined,and thermodynamically compatiblefirst-order hyperbolic systems.By construction,the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm.A very peculiar feature of our approach is that entropy is discretized directly,while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization.The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs,including both,conservative and non-conservative products,as well as potentially stiff algebraic relaxation source terms,provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law,such as the conservation of total energy density.The proposed family offinite volume schemes is based on the seminal work of Abgrall[1],where for thefirst time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented.We apply our new approach to three particular thermodynamically compatible systems:the equations of ideal magnetohydrodynamics(MHD)with thermodynamically compatible generalized Lagrangian multiplier(GLM)divergence cleaning,the unifiedfirst-order hyperbolic model of continuum mechanics proposed by Godunov,Peshkov,and Romenski(GPR model)and thefirst-order hyperbolic model for turbulent shallow waterflows of Gavrilyuk et al.In addition to formal mathematical proofs of the properties of our newfinite volume schemes,we also present a large set of numerical results in order to show their potential,efficiency,and practical applicability.展开更多
A novel numerical scheme to solve two coupled systems of conservation laws is introduced.The scheme is derived based on a relaxation approach and does not require information on the Lax curves of the coupled systems,w...A novel numerical scheme to solve two coupled systems of conservation laws is introduced.The scheme is derived based on a relaxation approach and does not require information on the Lax curves of the coupled systems,which simplifies the computation of suitable coupling data.The coupling condition for the underlying relaxation system plays a crucial role as it determines the behaviour of the scheme in the zero relaxation limit.The role of this condition is discussed,a consistency concept with respect to the original problem is introduced,the well-posedness is analyzed and explicit,nodal Riemann solvers are provided.Based on a case study considering the p-system of gas dynamics,a strategy for the design of the relaxation coupling condition within the new scheme is provided.展开更多
Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of ...Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy.This makes them quite attractive for several science and engineering applications.But,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative products.In this paper,we perform such an extension which improves the domain of the applicability of such schemes.The extension is carried out by writing the scheme in fluctuation form.We use the HLLI Riemann solver of Dumbser and Balsara(J.Comput.Phys.304:275-319,2016)as a building block for carrying out this extension.Because of the use of an HLL building block,the resulting scheme has a proper supersonic limit.The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the applicability.Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions,making it very easy for users to transition over to the present formulation.For conservation laws,the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO,with two major advantages:(i)It can capture jumps in stationary linearly degenerate wave families exactly.(i)It only requires the reconstruction to be applied once.Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows.Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow,multiphase debris flow and twolayer shallow water equations are also shown to document the robustness of the method.For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results.Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.展开更多
In this paper,we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations(PDEs)with uncertainties.The new approach is realized in the semi-discrete finite-volume fram...In this paper,we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations(PDEs)with uncertainties.The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essentially non-oscillatory(WENO)interpolations in(multidimensional)random space combined with second-order piecewise linear reconstruction in physical space.Compared with spectral approximations in the random space,the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy.The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations.In the latter case,the methods are also proven to be well-balanced and positivity-preserving.展开更多
This article considers Cauchy problem u(t) - (uv)(x) = 0, v(t) - u(x) = 0, u(x, 0) = u(0) (x) > 0, v(x, 0) = v(0)(x). A necessary and sufficient condition in guaranteeing that Cauchy problem admits a global C-1-sol...This article considers Cauchy problem u(t) - (uv)(x) = 0, v(t) - u(x) = 0, u(x, 0) = u(0) (x) > 0, v(x, 0) = v(0)(x). A necessary and sufficient condition in guaranteeing that Cauchy problem admits a global C-1-solution on t greater than or equal to 0 is obtained.展开更多
In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {（t, x） ｜ t ≥ 0, x ≥0} is considered. A suffic...In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {（t, x） ｜ t ≥ 0, x ≥0} is considered. A sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.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
The problem of the presence of Cantor part in the derivative of a solution to a hyperbolic system of conservation laws is considered. An overview of the techniques involved in the proof is given, and a collection of r...The problem of the presence of Cantor part in the derivative of a solution to a hyperbolic system of conservation laws is considered. An overview of the techniques involved in the proof is given, and a collection of related problems concludes the paper.展开更多
We consider first order quasilinear hyperbolic systems with vertical characteristics. It was shown in [4] that such systems can be exactly controllable with the help of internal controls applied to the equations corr...We consider first order quasilinear hyperbolic systems with vertical characteristics. It was shown in [4] that such systems can be exactly controllable with the help of internal controls applied to the equations corresponding to zero eigenvalues. However, it is possible that, for physical or engineering reasons, we can not put any control on the equations corresponding to zero eigenvalues. In this paper, we will establish the exact controllability only by means of physically meaningfnl internal controls applied to the equations corresponding to non-zero eigenvalues. We also show the exact controllability for a very simplified model by means of switching controls.展开更多
In this article, first, the authors prove that there exists a unique global smooth solution for the Cauthy problem to the hyperbolic conservation laws systems with relaxation; second, in the large time station, they p...In this article, first, the authors prove that there exists a unique global smooth solution for the Cauthy problem to the hyperbolic conservation laws systems with relaxation; second, in the large time station, they prove that the global smooth solutions of the hyperbolic conservation laws systems with relaxation converge to rarefaction waves solution at a determined L^P（p ≥ 2） decay rate.展开更多
A convergence theorem for the method of artificial viscosity applied to the nonstrictly hyperbolic system u(t)+1/2(3u2+v2)x=0, v(t)+(uv)x=0 is established. Convergence of a subsequence in the strong topology is proved...A convergence theorem for the method of artificial viscosity applied to the nonstrictly hyperbolic system u(t)+1/2(3u2+v2)x=0, v(t)+(uv)x=0 is established. Convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves.展开更多
This paper studies the nonlinear mixed problem for a class of symmetric hyperbolic systems with the boundary condition satisfying the dissipative condition about discontinuous data in higher dimension spaces, establis...This paper studies the nonlinear mixed problem for a class of symmetric hyperbolic systems with the boundary condition satisfying the dissipative condition about discontinuous data in higher dimension spaces, establishes the local existence theorem by using the method of a prior estimates, and obtains the structure of singularities of the solutions of such problems.展开更多
In this study, boundary control problems with Neumann conditions for 2 × 2 cooperative hyperbolic systems involving infinite order operators are considered. The existence and uniqueness of the states of these sys...In this study, boundary control problems with Neumann conditions for 2 × 2 cooperative hyperbolic systems involving infinite order operators are considered. The existence and uniqueness of the states of these systems are proved, and the formulation of the control problem for different observation functions is discussed.展开更多
In this paper, we consider cooperative hyperbolic systems involving Schr?dinger operator defined on ?Rn. First we prove the existence and uniqueness of the state for these systems. Then we find the necessary and suffi...In this paper, we consider cooperative hyperbolic systems involving Schr?dinger operator defined on ?Rn. First we prove the existence and uniqueness of the state for these systems. Then we find the necessary and sufficient conditions of optimal control for such systems of the boundary type. We also find the necessary and sufficient conditions of optimal control for same systems when the observation is on the boundary.展开更多
We are concerned with the derivation and analysis of one-dimensional hyperbolic systems of conservation laws modelling fluid flows such as the blood flow through compliant axisyminetric vessels. Early models derived a...We are concerned with the derivation and analysis of one-dimensional hyperbolic systems of conservation laws modelling fluid flows such as the blood flow through compliant axisyminetric vessels. Early models derived are nonconservative and/or nonho- mogeneous with measure source terms, which are endowed with infinitely many Riemann solutions for some Riemann data. In this paper, we derive a one-dimensional hyperbolic system that is conservative and homogeneous. Moreover, there exists a unique global Riemann solution for the Riemann problem for two vessels with arbitrarily large Riemann data, under a natural stability entropy criterion. The Riemann solutions may consist of four waves for some cases. The system can also be written as a 3 × 3 system for which strict hyperbolicity fails and the standing waves can be regarded as the contact discontinuities corresponding to the second family with zero eigenvalue.展开更多
In this article, we give the existence of global L^(∞)bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of Le Roux type. The main difficulty lies in establishing some compa...In this article, we give the existence of global L^(∞)bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of Le Roux type. The main difficulty lies in establishing some compactness estimates of the viscosity solutions because the system has been generalized from 2×2 to n×n and more linearly degenerate characteristic fields emerged, and the emergence of singularity in the region {v1= 0} is another difficulty.We obtain the existence of the global weak solutions using the compensated compactness method coupled with the construction of entropy-entropy flux and BV estimates on viscous solutions.展开更多
In this paper, we investigate a class of mixed initial-boundary value problems for a kind of n × n quasilinear hyperbolic systems of conservation laws on the quarter plan. We show that the structure of the pieeew...In this paper, we investigate a class of mixed initial-boundary value problems for a kind of n × n quasilinear hyperbolic systems of conservation laws on the quarter plan. We show that the structure of the pieeewise C^1 solution u = u（t, x） of the problem, which can be regarded as a perturbation of the corresponding Riemann problem, is globally similar to that of the solution u = U（x/t） of the corresponding Riemann problem. The piecewise C^1 solution u = u（t, x） to this kind of problems is globally structure-stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.展开更多
The paper aims to extend the notion of regional observability of the gradient to the semilinear hyperbolic case, in order to reconstruct the gradient of the initial conditions in a subregion w of the domain evolution ...The paper aims to extend the notion of regional observability of the gradient to the semilinear hyperbolic case, in order to reconstruct the gradient of the initial conditions in a subregion w of the domain evolution Ω. We start with an asymptotically linear system, the approach is based on an extension of the Hilbert uniqueness method （HUM） and Schauder＇s fixed point theorem. The analysis leads to an algorithm which is successfully numerically implemented and illustrated with examples and simulations.展开更多
This paper deals with absorbing boundary conditions for hyperbolic systems in one and two space dimensions.We prove the strict well-posedness of the resulting initial boundary value problem in 1D.Afterwards we establi...This paper deals with absorbing boundary conditions for hyperbolic systems in one and two space dimensions.We prove the strict well-posedness of the resulting initial boundary value problem in 1D.Afterwards we establish the GKS-stability of the corresponding Lax-Wendroff-type finite difference scheme.Hereby,we have to extend the classical proofs,since the(discretized) absorbing boundary conditions do not fit the standard form of boundary conditions for hyperbolic systems.展开更多
基金supported by the National Natural Science Foundation of China(12371217)the Fundamental Research Funds for the Central Universities(2232022D-27).
文摘For multidimensional first order semilinear hyperbolic systems of diagonal form without self-interaction,we show the global nonlinear stability of traveling wave solutions.
文摘In this paper,a new efficient,and at the same time,very simple and general class of thermodynamically compatiblefinite volume schemes is introduced for the discretization of nonlinear,overdetermined,and thermodynamically compatiblefirst-order hyperbolic systems.By construction,the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm.A very peculiar feature of our approach is that entropy is discretized directly,while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization.The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs,including both,conservative and non-conservative products,as well as potentially stiff algebraic relaxation source terms,provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law,such as the conservation of total energy density.The proposed family offinite volume schemes is based on the seminal work of Abgrall[1],where for thefirst time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented.We apply our new approach to three particular thermodynamically compatible systems:the equations of ideal magnetohydrodynamics(MHD)with thermodynamically compatible generalized Lagrangian multiplier(GLM)divergence cleaning,the unifiedfirst-order hyperbolic model of continuum mechanics proposed by Godunov,Peshkov,and Romenski(GPR model)and thefirst-order hyperbolic model for turbulent shallow waterflows of Gavrilyuk et al.In addition to formal mathematical proofs of the properties of our newfinite volume schemes,we also present a large set of numerical results in order to show their potential,efficiency,and practical applicability.
基金Funding Open Access funding enabled and organized by Projekt DEALthe Deutsche Forschungsgemeinschaft(DFG,German Research Foundation)for the financial support through 320021702/GRK2326,333849990/IRTG-2379,B04,B05,and B06 of 442047500/SFB1481,HE5386/18-1,19-2,22-1,23-1,25-1,ERS SFDdM035 and under Germany’s Excellence Strategy EXC-2023 Internet of Production 390621612 and under the Excellence Strategy of the Federal Government and the Länder.Support through the EU DATAHYKING is also acknowledged.
文摘A novel numerical scheme to solve two coupled systems of conservation laws is introduced.The scheme is derived based on a relaxation approach and does not require information on the Lax curves of the coupled systems,which simplifies the computation of suitable coupling data.The coupling condition for the underlying relaxation system plays a crucial role as it determines the behaviour of the scheme in the zero relaxation limit.The role of this condition is discussed,a consistency concept with respect to the original problem is introduced,the well-posedness is analyzed and explicit,nodal Riemann solvers are provided.Based on a case study considering the p-system of gas dynamics,a strategy for the design of the relaxation coupling condition within the new scheme is provided.
基金support via NSF grants NSF-19-04774,NSF-AST-2009776,NASA-2020-1241NASA grant 80NSSC22K0628.DSB+3 种基金HK acknowledge support from a Vajra award,VJR/2018/00129a travel grant from Notre Dame Internationalsupport via AFOSR grant FA9550-20-1-0055NSF grant DMS-2010107.
文摘Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy.This makes them quite attractive for several science and engineering applications.But,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative products.In this paper,we perform such an extension which improves the domain of the applicability of such schemes.The extension is carried out by writing the scheme in fluctuation form.We use the HLLI Riemann solver of Dumbser and Balsara(J.Comput.Phys.304:275-319,2016)as a building block for carrying out this extension.Because of the use of an HLL building block,the resulting scheme has a proper supersonic limit.The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the applicability.Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions,making it very easy for users to transition over to the present formulation.For conservation laws,the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO,with two major advantages:(i)It can capture jumps in stationary linearly degenerate wave families exactly.(i)It only requires the reconstruction to be applied once.Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows.Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow,multiphase debris flow and twolayer shallow water equations are also shown to document the robustness of the method.For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results.Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.
基金supported in part by the NSF grant DMS-2208438.The work of M.Herty was supported in part by the DFG(German Research Foundation)through 20021702/GRK2326,333849990/IRTG-2379,HE5386/18-1,19-2,22-1,23-1under Germany’s Excellence Strategy EXC-2023 Internet of Production 390621612+1 种基金The work of A.Kurganov was supported in part by the NSFC grant 12171226the fund of the Guangdong Provincial Key Laboratory of Computational Science and Material Design,China(No.2019B030301001).
文摘In this paper,we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations(PDEs)with uncertainties.The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essentially non-oscillatory(WENO)interpolations in(multidimensional)random space combined with second-order piecewise linear reconstruction in physical space.Compared with spectral approximations in the random space,the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy.The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations.In the latter case,the methods are also proven to be well-balanced and positivity-preserving.
基金Project supported by the NSF of Fujian Province (A97020)
文摘This article considers Cauchy problem u(t) - (uv)(x) = 0, v(t) - u(x) = 0, u(x, 0) = u(0) (x) > 0, v(x, 0) = v(0)(x). A necessary and sufficient condition in guaranteeing that Cauchy problem admits a global C-1-solution on t greater than or equal to 0 is obtained.
文摘In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {（t, x） ｜ t ≥ 0, x ≥0} is considered. A sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.
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.
文摘The problem of the presence of Cantor part in the derivative of a solution to a hyperbolic system of conservation laws is considered. An overview of the techniques involved in the proof is given, and a collection of related problems concludes the paper.
文摘We consider first order quasilinear hyperbolic systems with vertical characteristics. It was shown in [4] that such systems can be exactly controllable with the help of internal controls applied to the equations corresponding to zero eigenvalues. However, it is possible that, for physical or engineering reasons, we can not put any control on the equations corresponding to zero eigenvalues. In this paper, we will establish the exact controllability only by means of physically meaningfnl internal controls applied to the equations corresponding to non-zero eigenvalues. We also show the exact controllability for a very simplified model by means of switching controls.
基金This research is supported by "Foundation of office of overseas Chinese affair under the state council: 03QZR09"
文摘In this article, first, the authors prove that there exists a unique global smooth solution for the Cauthy problem to the hyperbolic conservation laws systems with relaxation; second, in the large time station, they prove that the global smooth solutions of the hyperbolic conservation laws systems with relaxation converge to rarefaction waves solution at a determined L^P（p ≥ 2） decay rate.
文摘A convergence theorem for the method of artificial viscosity applied to the nonstrictly hyperbolic system u(t)+1/2(3u2+v2)x=0, v(t)+(uv)x=0 is established. Convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves.
文摘This paper studies the nonlinear mixed problem for a class of symmetric hyperbolic systems with the boundary condition satisfying the dissipative condition about discontinuous data in higher dimension spaces, establishes the local existence theorem by using the method of a prior estimates, and obtains the structure of singularities of the solutions of such problems.
文摘In this study, boundary control problems with Neumann conditions for 2 × 2 cooperative hyperbolic systems involving infinite order operators are considered. The existence and uniqueness of the states of these systems are proved, and the formulation of the control problem for different observation functions is discussed.
文摘In this paper, we consider cooperative hyperbolic systems involving Schr?dinger operator defined on ?Rn. First we prove the existence and uniqueness of the state for these systems. Then we find the necessary and sufficient conditions of optimal control for such systems of the boundary type. We also find the necessary and sufficient conditions of optimal control for same systems when the observation is on the boundary.
基金supported in part by the National Science Foundation under Grants DMS-0935967the National Science Foundation under Grants DMS-0807551+2 种基金the National Science Foundation under Grants DMS-0720925the National Science Foundation under Grants DMS-0505473the Natural Science Foundation of China under Grant NSFC-10728101,and the Royal Society-Wolfson Research Merit Award (UK)
文摘We are concerned with the derivation and analysis of one-dimensional hyperbolic systems of conservation laws modelling fluid flows such as the blood flow through compliant axisyminetric vessels. Early models derived are nonconservative and/or nonho- mogeneous with measure source terms, which are endowed with infinitely many Riemann solutions for some Riemann data. In this paper, we derive a one-dimensional hyperbolic system that is conservative and homogeneous. Moreover, there exists a unique global Riemann solution for the Riemann problem for two vessels with arbitrarily large Riemann data, under a natural stability entropy criterion. The Riemann solutions may consist of four waves for some cases. The system can also be written as a 3 × 3 system for which strict hyperbolicity fails and the standing waves can be regarded as the contact discontinuities corresponding to the second family with zero eigenvalue.
基金supported by the National Science Foundation of China(11572148,11671193)the National Research Foundation for the Doctoral Program of Higher Education of China(20133218110025)
文摘In this article, we give the existence of global L^(∞)bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of Le Roux type. The main difficulty lies in establishing some compactness estimates of the viscosity solutions because the system has been generalized from 2×2 to n×n and more linearly degenerate characteristic fields emerged, and the emergence of singularity in the region {v1= 0} is another difficulty.We obtain the existence of the global weak solutions using the compensated compactness method coupled with the construction of entropy-entropy flux and BV estimates on viscous solutions.
基金supported by the National Natural Science Foundation of China under Grant No.10671124
文摘In this paper, we investigate a class of mixed initial-boundary value problems for a kind of n × n quasilinear hyperbolic systems of conservation laws on the quarter plan. We show that the structure of the pieeewise C^1 solution u = u（t, x） of the problem, which can be regarded as a perturbation of the corresponding Riemann problem, is globally similar to that of the solution u = U（x/t） of the corresponding Riemann problem. The piecewise C^1 solution u = u（t, x） to this kind of problems is globally structure-stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.
文摘The paper aims to extend the notion of regional observability of the gradient to the semilinear hyperbolic case, in order to reconstruct the gradient of the initial conditions in a subregion w of the domain evolution Ω. We start with an asymptotically linear system, the approach is based on an extension of the Hilbert uniqueness method （HUM） and Schauder＇s fixed point theorem. The analysis leads to an algorithm which is successfully numerically implemented and illustrated with examples and simulations.
文摘This paper deals with absorbing boundary conditions for hyperbolic systems in one and two space dimensions.We prove the strict well-posedness of the resulting initial boundary value problem in 1D.Afterwards we establish the GKS-stability of the corresponding Lax-Wendroff-type finite difference scheme.Hereby,we have to extend the classical proofs,since the(discretized) absorbing boundary conditions do not fit the standard form of boundary conditions for hyperbolic systems.
