In recent ten years high resolution difference schenies for the computation of thefull unsteady Eulerian system of equations for invisid compressible gas finds celebratedprogress. This paper tests furtherly, by a comp...In recent ten years high resolution difference schenies for the computation of thefull unsteady Eulerian system of equations for invisid compressible gas finds celebratedprogress. This paper tests furtherly, by a complex two-dimensional unsteady problem,four recent schemes. to them attentions are paid. The test problem is the initial stageof a two-dimensional diffraction and reflection of a plane shock wave, impinging on arectangular obstacle. At whose top side there are two sharp corners, near which flow.parameters finds severe variation. There is occurrence of expansion fan with a centerand also concentrated vortices. To simulate them well, the schemes should have goodadaptivity. The special shock Mach number M,=2.068 is so chosen, that at this M,the partical velocity behind impinging shock in fixed coordinate system is just equal tothe speed of sound there, this condition also occurs along a curve in the region ofexpansion fan with a center at the corner. This can clarify the computational featureof different schemes in case,when one of the eigenvalues is just zero. Zero eigenvaluemay spoil some schemes locally. Graphical visualization of the computational resultsmay, show features of the tested schemes about the shock wave resolution, schemeviscosity, expansion wave and the ability. to simulate the process of the generation ofunsteadv concentrated vortex.展开更多
In this paper we study a kind of mixed anti-diffusion method for partial differntial equations. Firstly, we use the method to construct some difference schemes for the conservation laws. The schemes are of second orde...In this paper we study a kind of mixed anti-diffusion method for partial differntial equations. Firstly, we use the method to construct some difference schemes for the conservation laws. The schemes are of second order accuracy and are total variation decreasing (TVD). In particular, there are only three knots involved in the schemes. Secondly, we extend the method to construct a few high accuracy difference schemes for elliptic and parabolic equations. Numerical experiments are carried out to illustrate the efficiency of the method.展开更多
Viscoelastic fluids due to their non-linear nature play an important role in process and polymer industries. These non-linear characteristics of fluid, influence final outcome of the product. Such processes though loo...Viscoelastic fluids due to their non-linear nature play an important role in process and polymer industries. These non-linear characteristics of fluid, influence final outcome of the product. Such processes though look simple are numerically challenging to study, due to the loss of numerical stability. Over the years, various methodologies have been developed to overcome this numerical limitation. In spite of this, numerical solutions are considered distant from accuracy, as first-order upwind-differencing scheme （UDS） is often employed for improving the stability of algorithm. To elude this effect, some works been reported in the past, where high-resolution-schemes （HRS） were employed and Deborah number was varied. However, these works are limited to creeping flows and do not detail any information on the numerical stability of HRS. Hence, this article presents the numerical study of high shearing contraction flows, where stability of HRS are addressed in reference to fluid elasticity. Results suggest that all I-IRS show some order of undue oscillations in flow variable profiles, measured along vertical lines placed near contraction region in the upstream section of domain, at varied elasticity number E ~ 5. Furthermore, by E, a clear relationship between numerical stability of HRS and E was obtained, which states that the order of undue oscillations in flow variable profiles is directly proportional to E.展开更多
Presents a method of proof which improves the estimates of entropy production for general total variation diminishing (TVD) schemes. Elements of the general theory of TVD schemes; Basis for obtaining the entropy inequ...Presents a method of proof which improves the estimates of entropy production for general total variation diminishing (TVD) schemes. Elements of the general theory of TVD schemes; Basis for obtaining the entropy inequality of a class of second order resolution-TVD schemes for strict convex conservation laws; Definition of discrete entropy inequality.展开更多
This paper presents a class of high resolution local time step schemes for nonlinear hyperbolic conservation laws and the closely related convection-diffusion equations, by projecting the solution increments of the un...This paper presents a class of high resolution local time step schemes for nonlinear hyperbolic conservation laws and the closely related convection-diffusion equations, by projecting the solution increments of the underlying partial differential equations （PDE） at each local time step. The main advantages are that they are of good consistency, and it is convenient to implement them. The schemes are L^∞ stable, satisfy a cell entropy inequality, and may be extended to the initial boundary value problem of general unsteady PDEs with higher-order spatial derivatives. The high resolution schemes are given by combining the reconstruction technique with a second order TVD Runge-Kutta scheme or a Lax-Wendroff type method, respectively. The schemes are used to solve a linear convection-diffusion equation, the nonlinear inviscid Burgers＇ equation, the one- and two-dimensional compressible Euler equations, and the two-dimensional incompressible Navier-Stokes equations. The numerical results show that the schemes are of higher-order accuracy, and efficient in saving computational cost, especially, for the case of combining the present schemes with the adaptive mesh method [15]. The correct locations of the slow moving or stronger discontinuities are also obtained, although the schemes are slightly nonconservative.展开更多
In this study,fully coupled thermo-poroelastic saturated media are simulated by a grid/cell adaptive central high resolution scheme.The central method corresponds to the second order Kurganov-Tadmor(KT)scheme working ...In this study,fully coupled thermo-poroelastic saturated media are simulated by a grid/cell adaptive central high resolution scheme.The central method corresponds to the second order Kurganov-Tadmor(KT)scheme working on adapted cells with the total variation diminishing(TVD)stability condition.The coupled equations include motion,fluid flow,heat flow,continuity condition,and a constitutive equation.The grid/cell adaptation is performed by the interpolating wavelet transform in the multiresolution framework to capture fine scale responses and to obtain a computationally effective solver.With respect to the use of central schemes,the coupled equations should be re-expressed as a system of coupled first-order hyperbolic-parabolic partial differential equations(PDEs)with possible source(load)terms.The system is initially derived in the Cartesian coordinate system,and it is subsequently modified to consider a spherical cavity in isotropic,symmetric,and saturated media in the spherical coordinate system.It is assumed that the cavity boundary is subjected to sudden time-dependent thermal/mechanical sources.Discontinuous propagating fronts develop in the media due to the aforementioned loading.It is challenging to handle these solutions with numerical methods,and special attention is required to prevent/control numerical dispersion and dissipation.Hence,as previously mentioned,adaptive central high resolution schemes are employed in the present study.展开更多
In this work,we have developed a fifth-order alternative mapped weighted essentially nonoscillatory(AWENO-M)finite volume scheme using non-linear weights of mapped WENO reconstruction scheme of Henrick et al.(J.Comput...In this work,we have developed a fifth-order alternative mapped weighted essentially nonoscillatory(AWENO-M)finite volume scheme using non-linear weights of mapped WENO reconstruction scheme of Henrick et al.(J.Comput.Phys.,207(2005),pp.542-567)for solving hyperbolic conservation laws.The reconstruction of numerical flux is done using primitive variables instead of conservative variables.The present scheme results in less spurious oscillations near discontinuities and shows higher-order accuracy at critical points compared to the alternative WENO scheme(AWENO)based on traditional non-linear weights of Jiang and Shu(J.Comput.Phys.,228(1996),pp.202-228).The third-order Runge-Kutta method has been used for solution advancement in time.The Harten-Lax-van Leer-Contact(HLLC)shock-capturing method is used to provide necessary upwinding into the solution.The performance of the present scheme is evaluated in terms of accuracy,computational cost,and resolution of discontinuities by using various one and two-dimensional test cases.展开更多
This paper is concerned with a new version of the Osher-Solomon Riemann solver and is based on a numerical integration of the path-dependent dissipation matrix.The resulting scheme is much simpler than the original on...This paper is concerned with a new version of the Osher-Solomon Riemann solver and is based on a numerical integration of the path-dependent dissipation matrix.The resulting scheme is much simpler than the original one and is applicable to general hyperbolic conservation laws,while retaining the attractive features of the original solver:the method is entropy-satisfying,differentiable and complete in the sense that it attributes a different numerical viscosity to each characteristic field,in particular to the intermediate ones,since the full eigenstructure of the underlying hyperbolic system is used.To illustrate the potential of the proposed scheme we show applications to the following hyperbolic conservation laws:Euler equations of compressible gasdynamics with ideal gas and real gas equation of state,classical and relativistic MHD equations as well as the equations of nonlinear elasticity.To the knowledge of the authors,apart from the Euler equations with ideal gas,an Osher-type scheme has never been devised before for any of these complicated PDE systems.Since our new general Riemann solver can be directly used as a building block of high order finite volume and discontinuous Galerkin schemes we also show the extension to higher order of accuracy and multiple space dimensions in the new framework of PNPM schemes on unstructured meshes recently proposed in[9].展开更多
The level set method, TVD scheme of second order upwind procedure coupled with flux limiter, ENO velocity extension procedure inside the huhble, and MAC projection algorithm were incorporated to simulate the whole col...The level set method, TVD scheme of second order upwind procedure coupled with flux limiter, ENO velocity extension procedure inside the huhble, and MAC projection algorithm were incorporated to simulate the whole collapse e volution of a cavitation bubble near a rigid wall with many complicated phenomena, such as topology distortion and shrinking, jet impact, bubble breaking into a toroidal form, and diminishing volume to zero, etc. The huhhle shape, evolution and distribution of velocity and pressure fields of the fluid during the bubble collapsing were investigated. It is found that the method is numerically stable and has good convergence property, and the results are in good agreements with those in previous work.展开更多
文摘In recent ten years high resolution difference schenies for the computation of thefull unsteady Eulerian system of equations for invisid compressible gas finds celebratedprogress. This paper tests furtherly, by a complex two-dimensional unsteady problem,four recent schemes. to them attentions are paid. The test problem is the initial stageof a two-dimensional diffraction and reflection of a plane shock wave, impinging on arectangular obstacle. At whose top side there are two sharp corners, near which flow.parameters finds severe variation. There is occurrence of expansion fan with a centerand also concentrated vortices. To simulate them well, the schemes should have goodadaptivity. The special shock Mach number M,=2.068 is so chosen, that at this M,the partical velocity behind impinging shock in fixed coordinate system is just equal tothe speed of sound there, this condition also occurs along a curve in the region ofexpansion fan with a center at the corner. This can clarify the computational featureof different schemes in case,when one of the eigenvalues is just zero. Zero eigenvaluemay spoil some schemes locally. Graphical visualization of the computational resultsmay, show features of the tested schemes about the shock wave resolution, schemeviscosity, expansion wave and the ability. to simulate the process of the generation ofunsteadv concentrated vortex.
文摘In this paper we study a kind of mixed anti-diffusion method for partial differntial equations. Firstly, we use the method to construct some difference schemes for the conservation laws. The schemes are of second order accuracy and are total variation decreasing (TVD). In particular, there are only three knots involved in the schemes. Secondly, we extend the method to construct a few high accuracy difference schemes for elliptic and parabolic equations. Numerical experiments are carried out to illustrate the efficiency of the method.
文摘Viscoelastic fluids due to their non-linear nature play an important role in process and polymer industries. These non-linear characteristics of fluid, influence final outcome of the product. Such processes though look simple are numerically challenging to study, due to the loss of numerical stability. Over the years, various methodologies have been developed to overcome this numerical limitation. In spite of this, numerical solutions are considered distant from accuracy, as first-order upwind-differencing scheme （UDS） is often employed for improving the stability of algorithm. To elude this effect, some works been reported in the past, where high-resolution-schemes （HRS） were employed and Deborah number was varied. However, these works are limited to creeping flows and do not detail any information on the numerical stability of HRS. Hence, this article presents the numerical study of high shearing contraction flows, where stability of HRS are addressed in reference to fluid elasticity. Results suggest that all I-IRS show some order of undue oscillations in flow variable profiles, measured along vertical lines placed near contraction region in the upstream section of domain, at varied elasticity number E ~ 5. Furthermore, by E, a clear relationship between numerical stability of HRS and E was obtained, which states that the order of undue oscillations in flow variable profiles is directly proportional to E.
基金The project supported partly by National Natural Science Foundation No.19901031, State Major KeyProject for Basic Research.
文摘Presents a method of proof which improves the estimates of entropy production for general total variation diminishing (TVD) schemes. Elements of the general theory of TVD schemes; Basis for obtaining the entropy inequality of a class of second order resolution-TVD schemes for strict convex conservation laws; Definition of discrete entropy inequality.
基金This research was partially sponsored by the National Basic Research Program under the Grant 2005CB321703, National Natural Science Foundation of China （No. 10431050, 10576001）, SRF for R0CS, SEM, the Alexander von Humboldt foundation, and the Deutsche Forschungsgemeinschaft （DFG Wa 633/10-3）.Acknowledgments. The authors thank Professor Tao Tang for numerous discussions during the preparation of this work, and also thank the referees for many helpful suggestions.
文摘This paper presents a class of high resolution local time step schemes for nonlinear hyperbolic conservation laws and the closely related convection-diffusion equations, by projecting the solution increments of the underlying partial differential equations （PDE） at each local time step. The main advantages are that they are of good consistency, and it is convenient to implement them. The schemes are L^∞ stable, satisfy a cell entropy inequality, and may be extended to the initial boundary value problem of general unsteady PDEs with higher-order spatial derivatives. The high resolution schemes are given by combining the reconstruction technique with a second order TVD Runge-Kutta scheme or a Lax-Wendroff type method, respectively. The schemes are used to solve a linear convection-diffusion equation, the nonlinear inviscid Burgers＇ equation, the one- and two-dimensional compressible Euler equations, and the two-dimensional incompressible Navier-Stokes equations. The numerical results show that the schemes are of higher-order accuracy, and efficient in saving computational cost, especially, for the case of combining the present schemes with the adaptive mesh method [15]. The correct locations of the slow moving or stronger discontinuities are also obtained, although the schemes are slightly nonconservative.
基金The authors gratefully acknowledge the financial support of Iran National Science Foundation(INSF).
文摘In this study,fully coupled thermo-poroelastic saturated media are simulated by a grid/cell adaptive central high resolution scheme.The central method corresponds to the second order Kurganov-Tadmor(KT)scheme working on adapted cells with the total variation diminishing(TVD)stability condition.The coupled equations include motion,fluid flow,heat flow,continuity condition,and a constitutive equation.The grid/cell adaptation is performed by the interpolating wavelet transform in the multiresolution framework to capture fine scale responses and to obtain a computationally effective solver.With respect to the use of central schemes,the coupled equations should be re-expressed as a system of coupled first-order hyperbolic-parabolic partial differential equations(PDEs)with possible source(load)terms.The system is initially derived in the Cartesian coordinate system,and it is subsequently modified to consider a spherical cavity in isotropic,symmetric,and saturated media in the spherical coordinate system.It is assumed that the cavity boundary is subjected to sudden time-dependent thermal/mechanical sources.Discontinuous propagating fronts develop in the media due to the aforementioned loading.It is challenging to handle these solutions with numerical methods,and special attention is required to prevent/control numerical dispersion and dissipation.Hence,as previously mentioned,adaptive central high resolution schemes are employed in the present study.
基金the computational facilities of CFD Lab.,MIED,IITRoorkee established with the FIST grant(DST-354-MID)from DST,Government of India.
文摘In this work,we have developed a fifth-order alternative mapped weighted essentially nonoscillatory(AWENO-M)finite volume scheme using non-linear weights of mapped WENO reconstruction scheme of Henrick et al.(J.Comput.Phys.,207(2005),pp.542-567)for solving hyperbolic conservation laws.The reconstruction of numerical flux is done using primitive variables instead of conservative variables.The present scheme results in less spurious oscillations near discontinuities and shows higher-order accuracy at critical points compared to the alternative WENO scheme(AWENO)based on traditional non-linear weights of Jiang and Shu(J.Comput.Phys.,228(1996),pp.202-228).The third-order Runge-Kutta method has been used for solution advancement in time.The Harten-Lax-van Leer-Contact(HLLC)shock-capturing method is used to provide necessary upwinding into the solution.The performance of the present scheme is evaluated in terms of accuracy,computational cost,and resolution of discontinuities by using various one and two-dimensional test cases.
基金financed by the Italian Ministry of Research(MIUR)under the project PRIN 2007 and by MIUR and the British Council under the project British-Italian Partnership Programme for young researchers 2008-2009。
文摘This paper is concerned with a new version of the Osher-Solomon Riemann solver and is based on a numerical integration of the path-dependent dissipation matrix.The resulting scheme is much simpler than the original one and is applicable to general hyperbolic conservation laws,while retaining the attractive features of the original solver:the method is entropy-satisfying,differentiable and complete in the sense that it attributes a different numerical viscosity to each characteristic field,in particular to the intermediate ones,since the full eigenstructure of the underlying hyperbolic system is used.To illustrate the potential of the proposed scheme we show applications to the following hyperbolic conservation laws:Euler equations of compressible gasdynamics with ideal gas and real gas equation of state,classical and relativistic MHD equations as well as the equations of nonlinear elasticity.To the knowledge of the authors,apart from the Euler equations with ideal gas,an Osher-type scheme has never been devised before for any of these complicated PDE systems.Since our new general Riemann solver can be directly used as a building block of high order finite volume and discontinuous Galerkin schemes we also show the extension to higher order of accuracy and multiple space dimensions in the new framework of PNPM schemes on unstructured meshes recently proposed in[9].
文摘The level set method, TVD scheme of second order upwind procedure coupled with flux limiter, ENO velocity extension procedure inside the huhble, and MAC projection algorithm were incorporated to simulate the whole collapse e volution of a cavitation bubble near a rigid wall with many complicated phenomena, such as topology distortion and shrinking, jet impact, bubble breaking into a toroidal form, and diminishing volume to zero, etc. The huhhle shape, evolution and distribution of velocity and pressure fields of the fluid during the bubble collapsing were investigated. It is found that the method is numerically stable and has good convergence property, and the results are in good agreements with those in previous work.
