We show how to combine in a natural way(i.e.,without any test nor switch)the conservative and non-conservative formulations of an hyperbolic system that has a conservative form.This is inspired from two different clas...We show how to combine in a natural way(i.e.,without any test nor switch)the conservative and non-conservative formulations of an hyperbolic system that has a conservative form.This is inspired from two different classes of schemes:the residual distribution one(Abgrall in Commun Appl Math Comput 2(3):341–368,2020),and the active flux formulations(Eyman and Roe in 49th AIAA Aerospace Science Meeting,2011;Eyman in active flux.PhD thesis,University of Michigan,2013;Helzel et al.in J Sci Comput 80(3):35–61,2019;Barsukow in J Sci Comput 86(1):paper No.3,34,2021;Roe in J Sci Comput 73:1094–1114,2017).The solution is globally continuous,and as in the active flux method,described by a combination of point values and average values.Unlike the“classical”active flux methods,the meaning of the point-wise and cell average degrees of freedom is different,and hence follow different forms of PDEs;it is a conservative version of the cell average,and a possibly non-conservative one for the points.This new class of scheme is proved to satisfy a Lax-Wendroff-like theorem.We also develop a method to perform nonlinear stability.We illustrate the behaviour on several benchmarks,some quite challenging.展开更多
The present paper deals with the numerical solution of a two-dimensional linear hyperbolic equation by using the element-free Galerkin （EFG） method which is based on the moving least-square approximation for the tes...The present paper deals with the numerical solution of a two-dimensional linear hyperbolic equation by using the element-free Galerkin （EFG） method which is based on the moving least-square approximation for the test and trial functions. A variational method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Compared with numerical methods based on mesh, the EFG method for hyperbolic problems needs only the scattered nodes instead of meshing the domain of the problem. It neither requires any element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. The effectiveness of the EFG method for two-dimensional hyperbolic problems is investigated by two numerical examples in this paper.展开更多
In this paper, we investigate the solutions of the hyperbolic Yamabe problem for the （1 ＋ n）-dimensional Minkowski space-time. More precisely speaking, for the case of n = 1, we derive a general solution of the hyp...In this paper, we investigate the solutions of the hyperbolic Yamabe problem for the （1 ＋ n）-dimensional Minkowski space-time. More precisely speaking, for the case of n = 1, we derive a general solution of the hyperbolic Yamabe problem; for the case of n =2, 3, we study the global existence and blowup phenomena of smooth solutions of the hyperbolic Yamabe problem; while for general multi-dimensional case n ≥ 2, we discuss the global existence and non-existence for a kind of exact solutions of the hyperbolic Yamabe problem.展开更多
In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the a...In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the asymptotic solution are given. Then an exponentially fitted difference scheme is developed in an equidistant mesh. Finally, uniform convergence in small parameter is proved in the sense of discrete energy norm.展开更多
In this paper we consider the initial-boundary value problem for a second order hyperbolic equation with initial jump. The bounds on the derivatives of the exact solution are given. Then a difference scheme is constru...In this paper we consider the initial-boundary value problem for a second order hyperbolic equation with initial jump. The bounds on the derivatives of the exact solution are given. Then a difference scheme is constructed on a non-uniform grid. Finally, uniform convergence of the difference solution is proved in the sense of the discrete energy norm.展开更多
We describe and review non oscillatory residual distribution schemes that are rather natural extension of high order finite volume schemes when a special emphasis is put on the structure of the computational stencil.W...We describe and review non oscillatory residual distribution schemes that are rather natural extension of high order finite volume schemes when a special emphasis is put on the structure of the computational stencil.We provide their connections with standard stabilized finite element and discontinuous Galerkin schemes,show that their are really non oscillatory.We also discuss the extension to these methods to parabolic problems.We also draw some research perspectives.展开更多
In this paper,we discuss the notion of discrete conservation for hyperbolic conservation laws.We introduce what we call fluctuation splitting schemes(or residual distribution,also RDS)and show through several examples...In this paper,we discuss the notion of discrete conservation for hyperbolic conservation laws.We introduce what we call fluctuation splitting schemes(or residual distribution,also RDS)and show through several examples how these schemes lead to new developments.In particular,we show that most,if not all,known schemes can be rephrased in flux form and also how to satisfy additional conservation laws.This review paper is built on Abgrall et al.(Computers and Fluids 169:10-22,2018),Abgrall and Tokareva(SIAM SISC 39(5):A2345-A2364,2017),Abgrall(J Sci Comput 73:461-494,2017),Abgrall(Methods Appl Math 18(3):327-351,2018a)and Abgrall(J Comput Phys 372,640--666,2018b).This paper is also a direct consequence of the work of Roe,in particular Deconinck et al.(Comput Fluids 22(2/3):215-222,1993)and Roe(J Comput Phys 43:357-372,1981)where the notion of conservation was first introduced.In[26],Roe mentioned the Hermes project and the role of Dassault Aviation.Bruno Stoufflet,Vice President R&D and advanced business of this company,proposed me to have a detailed look at Deconinck et al.(Comput Fluids 22(2/3):215-222,1993).To be honest,at the time,I did not understand anything,and this was the case for several years.I was lucky to work with Katherine Mer,who at the time was a postdoc,and is now research engineer at CEA.She helped me a lot in understanding the notion of conservation.The present contribution can be seen as the result of my understanding after many years of playing around with the notion of residual distribution schemes(or fluctuation-splitting schemes)introduced by Roe.展开更多
A boundary problem for the Klein-Gordon equation in the strip O≤t≤T is considered with the boundary condition:the initial state at t=O and the final state at t=T.It is proven that the problem admits of an infinite n...A boundary problem for the Klein-Gordon equation in the strip O≤t≤T is considered with the boundary condition:the initial state at t=O and the final state at t=T.It is proven that the problem admits of an infinite number of solutions.The same result holds for a generic 2nd order hyperbolic equation in 2-variables.Using the result for the wave operator in 3-space dimensions we give a method to reconstruct functions whose integral on all unit spheres in R~3 is a given function.展开更多
In this paper, we construct a local supersonic flow in a 3-dimensional axis-symmetry nozzle when a uniform supersonic flow inserts the throat. We apply the local existence theory of boundary value problem for quasilin...In this paper, we construct a local supersonic flow in a 3-dimensional axis-symmetry nozzle when a uniform supersonic flow inserts the throat. We apply the local existence theory of boundary value problem for quasilinear hyperbolic system to solve this problem. The boundary value condition is set in particular to guarantee the character number condition. By this trick, the theory in quasilinear hyperbolic system can be employed to a large range of the boundary value problem.展开更多
A simple one-dimensional 2 x 2 hyperbolic system is considered in the paper.The model contains a linear hyperbolic equation, as well as a hyperbolic equation ofwhich the coefficients are about the solution of the line...A simple one-dimensional 2 x 2 hyperbolic system is considered in the paper.The model contains a linear hyperbolic equation, as well as a hyperbolic equation ofwhich the coefficients are about the solution of the linear one. The exact solution ispresented and discussed, then numerical experiments are given by TVD (or MmB)type schemes for Riemann problems. From the results, we know that the solutionsdo have δ-waves for some suitable initial data.展开更多
Existing mapped WENO schemes can hardly prevent spurious oscillations while preserving high resolutions at long output times.We reveal in this paper the essential reason of such phenomena.It is actually caused by that...Existing mapped WENO schemes can hardly prevent spurious oscillations while preserving high resolutions at long output times.We reveal in this paper the essential reason of such phenomena.It is actually caused by that the mapping function in these schemes can not preserve the order of the nonlinear weights of the stencils.The nonlinear weights may be increased for non-smooth stencils and be decreased for smooth stencils.It is then indicated to require the set of mapping functions to be order-preserving in mapped WENO schemes.Therefore,we propose a new mapped WENO scheme with a set of mapping functions to be order-preserving which exhibits a remarkable advantage over the mapped WENO schemes in references.For long output time simulations of the one-dimensional linear advection equation,the new scheme has the capacity to attain high resolutions and avoid spurious oscillations near discontinuities meanwhile.In addition,for the two-dimensional Euler problems with strong shock waves,the new scheme can significantly reduce the numerical oscillations.展开更多
A new type offinite volume WENO schemes for hyperbolic problems was devised in[33]by introducing the order-preserving(OP)criterion.In this continuing work,we extend the OP criterion to the WENO-Z-type schemes.Wefirstl...A new type offinite volume WENO schemes for hyperbolic problems was devised in[33]by introducing the order-preserving(OP)criterion.In this continuing work,we extend the OP criterion to the WENO-Z-type schemes.Wefirstly rewrite the formulas of the Z-type weights in a uniform form from a mapping perspective inspired by extensive numerical observations.Accordingly,we build the concept of the locally order-preserving(LOP)mapping which is an extension of the order-preserving(OP)mapping and the resultant improved WENO-Z-type schemes are denoted as LOP-GMWENO-X.There are four major advantages of the LOP-GMWENO-X schemes superior to the existing WENO-Z-type schemes.Firstly,the new schemes can amend the serious drawback of the existing WENO-Z-type schemes that most of them suffer from either producing severe spurious oscillations or failing to obtain high resolutions in long calculations of hyperbolic problems with discontinuities.Secondly,they can maintain considerably high resolutions on solving problems with high-order critical points at long output times.Thirdly,they can obtain evidently higher resolution in the region with high-frequency but smooth waves.Finally,they can significantly decrease the post-shock oscillations for simulations of some 2D problems with strong shock waves.Extensive benchmark examples are conducted to illustrate these advantages.展开更多
In this paper,we describe a residual distribution(RD)method where,contrarily to“standard”this type schemes,the mesh is not necessarily conformal.It also allows to use discontinuous elements,contrarily to the“stand...In this paper,we describe a residual distribution(RD)method where,contrarily to“standard”this type schemes,the mesh is not necessarily conformal.It also allows to use discontinuous elements,contrarily to the“standard”case where continuous elements are requested.Moreover,if continuity is forced,the scheme is similar to the standard RD case.Hence,the situation becomes comparable with the Discontinuous Galerkin(DG)method,but it is simpler to implement than DG and has guaranteed L^(∞)bounds.We focus on the second-order case,but the method can be easily generalized to higher degree polynomials.展开更多
We contimle the work initiated in [1] （Second order nonlinear evolution inclusions I： Existence and relaxation results. Acta Mathematics Science, English Series, 21（5）, 977-996 （2005）） and study the structural ...We contimle the work initiated in [1] （Second order nonlinear evolution inclusions I： Existence and relaxation results. Acta Mathematics Science, English Series, 21（5）, 977-996 （2005）） and study the structural properties of the solution set of second order evolution inclusions which are defined in the analytic framework of the evolution triple. For the convex problem we show that the solution set is compact Rs, while for the nonconvex problem we show that it is path connected, Also we show that the solution set is closed only if the multivalued nonlinearity is convex valued. Finally we illustrate the results by considering a nonlinear hyperbolic problem with discontinuities.展开更多
This paper presents new results for strong solutions and their coincidence sets of the obstacle problem for linear hyperbolic operators of first order. An inequality similar to the LewyStampacchia ones for elliptic an...This paper presents new results for strong solutions and their coincidence sets of the obstacle problem for linear hyperbolic operators of first order. An inequality similar to the LewyStampacchia ones for elliptic and parabolic problems is shown. Under nondegeneracy conditions the stability of the coincidence set is shown with respect to the variation of the data and with respect to approximation by semilinear hyperbolic problems. These results are applied to the asymptotic stability of the evolution problem with respect to the stationary coercive problem with obstacle.展开更多
基金the author was partially funded by the SNF project 200020_175784.
文摘We show how to combine in a natural way(i.e.,without any test nor switch)the conservative and non-conservative formulations of an hyperbolic system that has a conservative form.This is inspired from two different classes of schemes:the residual distribution one(Abgrall in Commun Appl Math Comput 2(3):341–368,2020),and the active flux formulations(Eyman and Roe in 49th AIAA Aerospace Science Meeting,2011;Eyman in active flux.PhD thesis,University of Michigan,2013;Helzel et al.in J Sci Comput 80(3):35–61,2019;Barsukow in J Sci Comput 86(1):paper No.3,34,2021;Roe in J Sci Comput 73:1094–1114,2017).The solution is globally continuous,and as in the active flux method,described by a combination of point values and average values.Unlike the“classical”active flux methods,the meaning of the point-wise and cell average degrees of freedom is different,and hence follow different forms of PDEs;it is a conservative version of the cell average,and a possibly non-conservative one for the points.This new class of scheme is proved to satisfy a Lax-Wendroff-like theorem.We also develop a method to perform nonlinear stability.We illustrate the behaviour on several benchmarks,some quite challenging.
基金Project supported by the Natural Science Foundation of Ningbo, China (Grant Nos 2009A610014, 2009A610154, 2008A610020 and 2007A610050)
文摘The present paper deals with the numerical solution of a two-dimensional linear hyperbolic equation by using the element-free Galerkin （EFG） method which is based on the moving least-square approximation for the test and trial functions. A variational method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Compared with numerical methods based on mesh, the EFG method for hyperbolic problems needs only the scattered nodes instead of meshing the domain of the problem. It neither requires any element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. The effectiveness of the EFG method for two-dimensional hyperbolic problems is investigated by two numerical examples in this paper.
基金Supported by the NNSF of China(11271323,91330105)the Zhejiang Provincial Natural Science Foundation of China(LZ13A010002)
文摘In this paper, we investigate the solutions of the hyperbolic Yamabe problem for the （1 ＋ n）-dimensional Minkowski space-time. More precisely speaking, for the case of n = 1, we derive a general solution of the hyperbolic Yamabe problem; for the case of n =2, 3, we study the global existence and blowup phenomena of smooth solutions of the hyperbolic Yamabe problem; while for general multi-dimensional case n ≥ 2, we discuss the global existence and non-existence for a kind of exact solutions of the hyperbolic Yamabe problem.
文摘In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the asymptotic solution are given. Then an exponentially fitted difference scheme is developed in an equidistant mesh. Finally, uniform convergence in small parameter is proved in the sense of discrete energy norm.
文摘In this paper we consider the initial-boundary value problem for a second order hyperbolic equation with initial jump. The bounds on the derivatives of the exact solution are given. Then a difference scheme is constructed on a non-uniform grid. Finally, uniform convergence of the difference solution is proved in the sense of the discrete energy norm.
基金The author has been supported by the FP7 Advanced Grant#226316“ADDECCO”.The help of Mario Richiuto(INRIA)and A.Larat(now CNRS researcher at Ecole Centrale de Paris,France)are warmly acknowledged.
文摘We describe and review non oscillatory residual distribution schemes that are rather natural extension of high order finite volume schemes when a special emphasis is put on the structure of the computational stencil.We provide their connections with standard stabilized finite element and discontinuous Galerkin schemes,show that their are really non oscillatory.We also discuss the extension to these methods to parabolic problems.We also draw some research perspectives.
文摘In this paper,we discuss the notion of discrete conservation for hyperbolic conservation laws.We introduce what we call fluctuation splitting schemes(or residual distribution,also RDS)and show through several examples how these schemes lead to new developments.In particular,we show that most,if not all,known schemes can be rephrased in flux form and also how to satisfy additional conservation laws.This review paper is built on Abgrall et al.(Computers and Fluids 169:10-22,2018),Abgrall and Tokareva(SIAM SISC 39(5):A2345-A2364,2017),Abgrall(J Sci Comput 73:461-494,2017),Abgrall(Methods Appl Math 18(3):327-351,2018a)and Abgrall(J Comput Phys 372,640--666,2018b).This paper is also a direct consequence of the work of Roe,in particular Deconinck et al.(Comput Fluids 22(2/3):215-222,1993)and Roe(J Comput Phys 43:357-372,1981)where the notion of conservation was first introduced.In[26],Roe mentioned the Hermes project and the role of Dassault Aviation.Bruno Stoufflet,Vice President R&D and advanced business of this company,proposed me to have a detailed look at Deconinck et al.(Comput Fluids 22(2/3):215-222,1993).To be honest,at the time,I did not understand anything,and this was the case for several years.I was lucky to work with Katherine Mer,who at the time was a postdoc,and is now research engineer at CEA.She helped me a lot in understanding the notion of conservation.The present contribution can be seen as the result of my understanding after many years of playing around with the notion of residual distribution schemes(or fluctuation-splitting schemes)introduced by Roe.
文摘A boundary problem for the Klein-Gordon equation in the strip O≤t≤T is considered with the boundary condition:the initial state at t=O and the final state at t=T.It is proven that the problem admits of an infinite number of solutions.The same result holds for a generic 2nd order hyperbolic equation in 2-variables.Using the result for the wave operator in 3-space dimensions we give a method to reconstruct functions whose integral on all unit spheres in R~3 is a given function.
文摘In this paper, we construct a local supersonic flow in a 3-dimensional axis-symmetry nozzle when a uniform supersonic flow inserts the throat. We apply the local existence theory of boundary value problem for quasilinear hyperbolic system to solve this problem. The boundary value condition is set in particular to guarantee the character number condition. By this trick, the theory in quasilinear hyperbolic system can be employed to a large range of the boundary value problem.
文摘A simple one-dimensional 2 x 2 hyperbolic system is considered in the paper.The model contains a linear hyperbolic equation, as well as a hyperbolic equation ofwhich the coefficients are about the solution of the linear one. The exact solution ispresented and discussed, then numerical experiments are given by TVD (or MmB)type schemes for Riemann problems. From the results, we know that the solutionsdo have δ-waves for some suitable initial data.
文摘Existing mapped WENO schemes can hardly prevent spurious oscillations while preserving high resolutions at long output times.We reveal in this paper the essential reason of such phenomena.It is actually caused by that the mapping function in these schemes can not preserve the order of the nonlinear weights of the stencils.The nonlinear weights may be increased for non-smooth stencils and be decreased for smooth stencils.It is then indicated to require the set of mapping functions to be order-preserving in mapped WENO schemes.Therefore,we propose a new mapped WENO scheme with a set of mapping functions to be order-preserving which exhibits a remarkable advantage over the mapped WENO schemes in references.For long output time simulations of the one-dimensional linear advection equation,the new scheme has the capacity to attain high resolutions and avoid spurious oscillations near discontinuities meanwhile.In addition,for the two-dimensional Euler problems with strong shock waves,the new scheme can significantly reduce the numerical oscillations.
文摘A new type offinite volume WENO schemes for hyperbolic problems was devised in[33]by introducing the order-preserving(OP)criterion.In this continuing work,we extend the OP criterion to the WENO-Z-type schemes.Wefirstly rewrite the formulas of the Z-type weights in a uniform form from a mapping perspective inspired by extensive numerical observations.Accordingly,we build the concept of the locally order-preserving(LOP)mapping which is an extension of the order-preserving(OP)mapping and the resultant improved WENO-Z-type schemes are denoted as LOP-GMWENO-X.There are four major advantages of the LOP-GMWENO-X schemes superior to the existing WENO-Z-type schemes.Firstly,the new schemes can amend the serious drawback of the existing WENO-Z-type schemes that most of them suffer from either producing severe spurious oscillations or failing to obtain high resolutions in long calculations of hyperbolic problems with discontinuities.Secondly,they can maintain considerably high resolutions on solving problems with high-order critical points at long output times.Thirdly,they can obtain evidently higher resolution in the region with high-frequency but smooth waves.Finally,they can significantly decrease the post-shock oscillations for simulations of some 2D problems with strong shock waves.Extensive benchmark examples are conducted to illustrate these advantages.
基金This research has been done under a CNES grant,a FP6 STREP(ADIGMA,Contrat 30719)a FP7 ERC Advanced Grant(ADDECCO,contract 226316).
文摘In this paper,we describe a residual distribution(RD)method where,contrarily to“standard”this type schemes,the mesh is not necessarily conformal.It also allows to use discontinuous elements,contrarily to the“standard”case where continuous elements are requested.Moreover,if continuity is forced,the scheme is similar to the standard RD case.Hence,the situation becomes comparable with the Discontinuous Galerkin(DG)method,but it is simpler to implement than DG and has guaranteed L^(∞)bounds.We focus on the second-order case,but the method can be easily generalized to higher degree polynomials.
文摘We contimle the work initiated in [1] （Second order nonlinear evolution inclusions I： Existence and relaxation results. Acta Mathematics Science, English Series, 21（5）, 977-996 （2005）） and study the structural properties of the solution set of second order evolution inclusions which are defined in the analytic framework of the evolution triple. For the convex problem we show that the solution set is compact Rs, while for the nonconvex problem we show that it is path connected, Also we show that the solution set is closed only if the multivalued nonlinearity is convex valued. Finally we illustrate the results by considering a nonlinear hyperbolic problem with discontinuities.
基金Partially supported by the Project FCT-POCTI/34471/MAT/2000
文摘This paper presents new results for strong solutions and their coincidence sets of the obstacle problem for linear hyperbolic operators of first order. An inequality similar to the LewyStampacchia ones for elliptic and parabolic problems is shown. Under nondegeneracy conditions the stability of the coincidence set is shown with respect to the variation of the data and with respect to approximation by semilinear hyperbolic problems. These results are applied to the asymptotic stability of the evolution problem with respect to the stationary coercive problem with obstacle.
