In this article,we study the exhaustive analysis of nonlinear wave interactions for a 2×2 homogeneous system of quasilinear hyperbolic partial differential equations(PDEs)governing the macroscopic production.We u...In this article,we study the exhaustive analysis of nonlinear wave interactions for a 2×2 homogeneous system of quasilinear hyperbolic partial differential equations(PDEs)governing the macroscopic production.We use the hodograph transformation and differential constraints technique to obtain the exact solution of governing equations.Furthermore,we study the interaction between simple waves in detail through exact solution of general initial value problem.Finally,we discuss the all possible interaction of elementary waves using the solution of Riemann problem.展开更多
In this paper the Tricomi problem for a nonlinear mixed type equation is studied. The coefficients of the mixed type equation are discontinuous on the line, where the equation changes its type. The existence of soluti...In this paper the Tricomi problem for a nonlinear mixed type equation is studied. The coefficients of the mixed type equation are discontinuous on the line, where the equation changes its type. The existence of solution to this problem is proved. The method developed in this paper can be applied to study more difficult problems for nonlinear mixed type equations arising in gas dynamics.展开更多
This paper explores the properties and mathematical formulations of multidimensional simple waves,extending the well-established theory of onedimensional simple waves to higher dimensions.The study focuses on the conn...This paper explores the properties and mathematical formulations of multidimensional simple waves,extending the well-established theory of onedimensional simple waves to higher dimensions.The study focuses on the connection between simple waves and the Monge-Ampère equation,particularly in the context of gas dynamics and potential flows.Key aspects include the characterization of simple waves in unsteady and steady flows,the role of characteristic lines,and the application of Hodograph and Legendre transformations to derive solutions.The paper also addresses the challenges and open questions in extending simple wave theory to more complex systems,such as non-reducible systems,radiative heat transfer,and chemical reactions.The research highlights both theoretical advancements and practical applications,providing a foundation for future studies in this area.展开更多
The one-dimensional steep slope shallow water equations are used to model the dam-break flow down a uniform slope with arbitrary inclination, and analytical solutions are derived by the hodograph transformation and th...The one-dimensional steep slope shallow water equations are used to model the dam-break flow down a uniform slope with arbitrary inclination, and analytical solutions are derived by the hodograph transformation and the Riemann＇s method in terms of evaluated integrals. An implicit analytical solution is obtained to evaluate the spatio-temporal distributions of dam-break flood hydrographs along the slope. For convenience, the solution for representative wave profiles and velocity distributions is shown in charts. Comparing with the Dressler＇s solution and WES experimental data, the analytical solution is seen reasonable.展开更多
Riemann problems for the compressible Euler system in two space dimensions are complicated and difficult,but a viable alternative remains missing.The author lists merits of one-dimensional Riemann problems and compare...Riemann problems for the compressible Euler system in two space dimensions are complicated and difficult,but a viable alternative remains missing.The author lists merits of one-dimensional Riemann problems and compares them with those for the current two-dimensional Riemann problems,to illustrate their worthiness.Two-dimensional Riemann problems are approached via the methodology promoted by Andy Majda in the spirits of modern applied mathematics;that is,simplified model is built via asymptotic analysis,numerical simulation and theoretical analysis.A simplified model called the pressure gradient system is derived from the full Euler system via an asymptotic process.State-of-the-art numerical methods in numerical simulations are used to discern small-scale structures of the solutions,e.g.,semi-hyperbolic patches.Analytical methods are used to establish the validity of the structure revealed in the numerical simulation.The entire process,used in many of Majda's programs,is shown here for the two-dimensional Riemann problems for the compressible Euler systems of conservation laws.展开更多
We investigate a hyperbolic system of one-dimensional isothermal fluid with liquid-vapor phase transition.The refraction-reflection phenomena are intensively analyzed when elementary waves travel across the two-phase ...We investigate a hyperbolic system of one-dimensional isothermal fluid with liquid-vapor phase transition.The refraction-reflection phenomena are intensively analyzed when elementary waves travel across the two-phase interface.We apply the characteristic method and hodograph transform of Riemann to reduce the nonlinear PDEs to a concise form.Specially for the case of incident rarefaction wave,reduced linear equations are convenient to solve by Laplace transform.Then an integral formula in wave interaction region is derived in this paper,instead of the hypergeometric functions solutions for non-isothermal polytropic gases.It is also observed that when incident waves travel from the vapor phase to the liquid phase,the refracted waves must be accelerated and move forward.展开更多
基金Ministry of Human Resource Development,Government of India,for the institute fellowship(grant no.IIT/ACAD/PGS&R/F.II/2/14MA90J08)from IIT KharagpurSERB,DST,India(Ref.No.MTR/2019/001210)for its financial support through MATRICS grant。
文摘In this article,we study the exhaustive analysis of nonlinear wave interactions for a 2×2 homogeneous system of quasilinear hyperbolic partial differential equations(PDEs)governing the macroscopic production.We use the hodograph transformation and differential constraints technique to obtain the exact solution of governing equations.Furthermore,we study the interaction between simple waves in detail through exact solution of general initial value problem.Finally,we discuss the all possible interaction of elementary waves using the solution of Riemann problem.
基金supported by National Natural Science Foundation of China (10531020)the Doctorial Foundation of National Educational Ministry (20090071110002)
文摘In this paper the Tricomi problem for a nonlinear mixed type equation is studied. The coefficients of the mixed type equation are discontinuous on the line, where the equation changes its type. The existence of solution to this problem is proved. The method developed in this paper can be applied to study more difficult problems for nonlinear mixed type equations arising in gas dynamics.
基金supported by the NSF of Chongqing City(No.CSTB2023NSCQLZX0035)the National Natural Science Foundation of China(No.12571262)
文摘This paper explores the properties and mathematical formulations of multidimensional simple waves,extending the well-established theory of onedimensional simple waves to higher dimensions.The study focuses on the connection between simple waves and the Monge-Ampère equation,particularly in the context of gas dynamics and potential flows.Key aspects include the characterization of simple waves in unsteady and steady flows,the role of characteristic lines,and the application of Hodograph and Legendre transformations to derive solutions.The paper also addresses the challenges and open questions in extending simple wave theory to more complex systems,such as non-reducible systems,radiative heat transfer,and chemical reactions.The research highlights both theoretical advancements and practical applications,providing a foundation for future studies in this area.
基金Project supported by the Major Program of the Natural Science Foundation of China(Grant No.51079130)the Fujian Provincial Major Course of Hydraulic
文摘The one-dimensional steep slope shallow water equations are used to model the dam-break flow down a uniform slope with arbitrary inclination, and analytical solutions are derived by the hodograph transformation and the Riemann＇s method in terms of evaluated integrals. An implicit analytical solution is obtained to evaluate the spatio-temporal distributions of dam-break flood hydrographs along the slope. For convenience, the solution for representative wave profiles and velocity distributions is shown in charts. Comparing with the Dressler＇s solution and WES experimental data, the analytical solution is seen reasonable.
基金supported partially by the National Science Foundation (No.DMS-0603859)
文摘Riemann problems for the compressible Euler system in two space dimensions are complicated and difficult,but a viable alternative remains missing.The author lists merits of one-dimensional Riemann problems and compares them with those for the current two-dimensional Riemann problems,to illustrate their worthiness.Two-dimensional Riemann problems are approached via the methodology promoted by Andy Majda in the spirits of modern applied mathematics;that is,simplified model is built via asymptotic analysis,numerical simulation and theoretical analysis.A simplified model called the pressure gradient system is derived from the full Euler system via an asymptotic process.State-of-the-art numerical methods in numerical simulations are used to discern small-scale structures of the solutions,e.g.,semi-hyperbolic patches.Analytical methods are used to establish the validity of the structure revealed in the numerical simulation.The entire process,used in many of Majda's programs,is shown here for the two-dimensional Riemann problems for the compressible Euler systems of conservation laws.
基金Supported by the National Natural Science Foundation of China(No.11901475)China Postdoctoral Science Foundation(No.2019M653815XB)Chongqing Special Postdoctoral Science Foundation(No.XmT2018045)。
文摘We investigate a hyperbolic system of one-dimensional isothermal fluid with liquid-vapor phase transition.The refraction-reflection phenomena are intensively analyzed when elementary waves travel across the two-phase interface.We apply the characteristic method and hodograph transform of Riemann to reduce the nonlinear PDEs to a concise form.Specially for the case of incident rarefaction wave,reduced linear equations are convenient to solve by Laplace transform.Then an integral formula in wave interaction region is derived in this paper,instead of the hypergeometric functions solutions for non-isothermal polytropic gases.It is also observed that when incident waves travel from the vapor phase to the liquid phase,the refracted waves must be accelerated and move forward.
