jump discontinuities of semilinear strictly hyperbolic systems in two variables creation and propagation the creation and propagation of jump discontinuities in the solutions of semilinear strictly hyperbolic systems is studied in the case where the initial data has a discrete set xi i 1n of jump discontinuities lets be the smallest closed set which satisfies i s is a union of forward characteristics ii s contains all the forward characteristics from the points xi i 1n iii if two forward characteristics ins intersect then all forward characteristics from the point of intersection lie ins

hyperbolic conservation laws with relaxation the effect of relaxation is important in many physical situations it is present in the kinetic theory of gases elasticity with memory gas flow with thermo non equilibrium water waves etc the governing equations often take the form of hyperbolic conservation laws with lower order terms in this article we present and analyze a simple model of hyperbolic conservation laws with relaxation effects dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified stability criteria for diffusion waves expansion waves and traveling waves are found and justified nonlinearly time asymptotic expansion and the energy method are used in the analysis for dissipative waves the expansion is similar in spirit to the chapman enskog expansion in the kinetic theory for shock waves however a different approach is needed

asymptotic behavior of solutions to a hyperbolic system with relaxation and boundary effect in this paper we study the initial boundary value problem of the following hyperbolic system with relaxation

convergence analysis of a projection semi implicit coupling scheme for fluid structure interaction problems in this paper we provide a convergence analysis of a projection semi implicit scheme for the simulation of fluid structure systems involving an incompressible viscous fluid the error analysis is performed on a fully discretized linear coupled problem a finite element approximation and a semi implicit time stepping strategy are respectively used for space and time discretization the fluid is described by the stokes equations the structure by the classical linear elastodynamics equations linearized elasticity plate or shell models and all changes of geometry are neglected we derive an error estimate in finite time and we prove that the time discretization error for the coupling scheme is at least sqrt delta t finally some numerical experiments that confirm the theoretical analysis are presented

numerical analysis of a linearised fluid structure interaction problem this paper describes the numerical analysis of a time dependent linearised fluid structure interaction problems involving a very viscous fluid and an elastic shell in small displacements for simplicity all changes of geometry are neglected a single variational formulation is proposed for the whole problem and generic discretisation strategies are introduced independently on the fluid and on the structure more precisely the space approximation of the fluid problem is realized by standard mixed finite elements the shell is approximated by dkt finite elements and time derivatives are approximated either by midpoint rules or by backward difference formula using fundamental energy estimates on the continuous problem written in a proper functional space on its discrete equivalent and on an associated error evolution equation we can prove that the proposed variational problem is well posed and that its approximation in space and time converges with optimal order to the continuous solution

analysis of a fluid structure interaction problem recast in an optimal control setting fluid structure interaction fsi simulation presents many computational difficulties particularly when the densities of the fluid and structure are close a previous report p kuberry and h lee comput methods appl mech engrg 267 2013 pp 594 605 has suggested that recasting the fsi problem in the context of optimal control may significantly reduce computation time this paper introduces a neumann type control along with detailed analysis for the stability and existence of an optimal solution for a given time step the existence of lagrange multipliers is proved and an optimality system is derived a gradient based optimization algorithm is then presented with numerical results confirming its effectiveness in computing an accurate solution

zero relaxation limit to centered rarefaction waves for a rate type viscoelastic system abstract we study a rate type viscoelastic system proposed by i suliciu 1990 internat j engrg sci 28 827 841 which is a 3 3 hyperbolic system with relaxation as the relaxation time tends to zero this system convergences to the well known p system formally in the case where the initial data are the riemann data such that the corresponding solutions of the p system are centered rarefaction waves we show that if the wave strength is suitably small then the solution for the relaxation system exists globally in time and converges to the solution of the corresponding rarefaction waves uniformly as the relaxation time goes to zero except for an initial layer the jump discontinuities in the solutions are decaying exponentially fast as time tends to infinity

an unconditionally stable semi implicit fsi finite element method abstract the paper addresses the numerical solution of fluid structure interaction fsi problems involving incompressible viscous newtonian fluid and hyperelastic material a well known challenge in computing fsi systems is to provide an effective time marching algorithm which avoids numerical instabilities due to the loose coupling of fluid and structure motion on the fsi interface in this work we introduce a semi implicit finite element scheme for an arbitrary lagrangian eulerian formulation of the fluid structure interaction problem the approach strongly enforces the coupling conditions on the fluid structure interface but requires only a linear problem to be solved on each time step further we prove that the numerical solution to the fully discrete problem satisfies the correct energy balance and the stability estimate follows without any extra model simplifications or assumptions on the time step the analysis covers the cases of saint venant kirchhoff compressible and incompressible neo hookean materials results of several numerical experiments are included to illustrate the properties of the method and its applicability for the simulation of certain hemodynamic flows we also experiment with the enforcement of material incompressibility condition in the finite element method via an integral constraint or alternatively letting the poisson ratio in the compressible model to be close to 1 2 from these experiments conclusions are drawn concerning the accuracy of flow statistics prediction for incompressible vs nearly incompressible structure models

asymptotic stability of planar rarefaction waves for the relaxation approximation of conservation laws in several dimensions abstract this paper concerns the large time behavior toward planar rarefaction waves of solutions for the relaxation approximation of conservation laws in several dimensions it is shown that a planar rarefaction wave is nonlinear stable in the sense that it is an asymptotic attractor for the relaxation approximation of conservation laws

stabilization of explicit coupling in fluid structure interaction involving fluid incompressibility abstract in this work we propose a stabilized explicit coupling scheme for fluid structure interaction problems involving a viscous incompressible fluid the coupled discrete formulation is based on nitsche s method with a time penalty term giving l 2 control on the fluid pressure variations at the interface for a linear model problem we prove that the scheme is stable in the energy norm irrespectively of the so called added mass effect namely the fluid structure density ratio and the geometry of the domain numerical experiments in the linear and non linear case show that optimal time accuracy can be obtained by performing one defect correction iteration

existence and non existence of global smooth solutions for p system with relaxation abstract in this paper we consider the cauchy problem for p system with relaxation under the assumption that the relaxation time e is sufficiently small we prove the existence of the global smooth solution to the cauchy problem with c1 initial data provided the c0 norm of the derivative of the initial data is of the order of e here is a small positive constant on the other hand when the initial density has compact support but is not identically zero we prove the global regular solution for the cauchy problem does not exist

optimization based additive decomposition of weakly coercive problems with applications we present an abstract mathematical framework for an optimization based additive decomposition of a large class of variational problems into a collection of concurrent subproblems the framework replaces a given monolithic problem by an equivalent constrained optimization formulation in which the subproblems define the optimization constraints and the objective is to minimize the mismatch between their solutions the significance of this reformulation stems from the fact that one can solve the resulting optimality system by an iterative process involving only solutions of the subproblems consequently assuming that stable numerical methods and efficient solvers are available for every subproblem our reformulation leads to robust and efficient numerical algorithms for a given monolithic problem by breaking it into subproblems that can be handled more easily an application of the framework to the oseen equations illustrates its potential

first order system least squares fosls for coupled fluid elastic problems mathematical models for the mechanical coupling between a moving fluid and an elastic solid are inherently nonlinear because the shape of the eulerian fluid domain is not known a priori it is at least partially determined by the displacement of the elastic solid in this paper a first order system least squares finite element formulation is used to solve the nonlinear system of model equations using different iteration techniques including an approach where the equations are fully coupled and two other approaches in which the equations are decoupled the discrete linear system of equations is solved using an algebraic multigrid solver as a preconditioner for a conjugate gradient iteration the numerical results show that the approach is optimal in the sense that computational cost is proportional to the degrees of freedom the results also show that the choice of iteration method from fully coupled to fully decoupled does not significantly effect computational cost but it does influence the error in the solution

an effective fluid structure interaction formulation for vascular dynamics by generalized robin conditions in this work we focus on the modeling and numerical simulation of the fluid structure interaction mechanism in vascular dynamics we first propose a simple membrane model to describe the deformation of the arterial wall which is derived from the koiter shell equations and is applicable to an arbitrary geometry secondly we consider a reformulation of the fluid structure problem in which the newly derived membrane model thanks to its simplicity is embedded into the fluid equations and will appear as a generalized robin boundary condition the original problem is then reduced to the solution of subsequent fluid equations defined on a moving domain and may be achieved with a fluid solver only we also derive a stability estimate for the resulting numerical scheme finally we propose new outflow absorbing boundary conditions which are easy to implement and allow us to reduce significantly the spurious pressure wave reflections that typically appear in artificially truncated computational domains we present several numerical results showing the effectiveness of the proposed approaches

an exact algorithm for hierarchically well formulated subsets in second order polynomial regression variable selection in multiple regression requires identifying the best subset from among a set of candidate predictors in the case of polynomial regression the variable selection process can be further complicated by the need to obtain subsets that are hierarchically well formulated we present a branch and bound algorithm for selection of the best hierarchically well formulated subset in second order polynomial regression we apply the new algorithm to a well known data set from the regression literature and compare the results with those obtained from a branch and bound algorithm that does not impose the hierarchical constraints this comparison reveals that the hierarchical constraints yield only a small penalty in explained variation we offer fortran and matlab implementations of the branch and bound algorithms as supplemental materials associated with this work

hyperbolic conservation laws with stiff relaxation terms and entropy we study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms reduced systems inviscid and viscous local conservation laws and weakly nonlinear limits are derived through asymptotic expansions an entropy condition is introduced for n n systems that ensures the hyperbolicity of the reduced inviscid system the resulting characteristic speeds are shown to be interlaced with those of the original system moreover the first correction to the reduced system is shown to be dissipative a partial converse is proved for 2 2 systems this structure is then applied to study the convergence to the reduced dynamics for the 2 2 case 1994 john wiley sons inc

zero dissipation limit to rarefaction waves for the one dimensional navier stokes equations of compressible isentropic gases we study the zero dissipation limit problem for the one dimensional navier stokes equations of compressible isentropic gases in the case that the corresponding euler equations have rarefaction wave solutions we prove that the solutions of the navier stokes equations with centered rarefaction wave data exist for all time and converge to the centered rarefaction waves as the viscosity vanishes uniformly away from the initial discontinuities in the case that either the effects of initial layers are ignored or the rarefaction waves are smooth we then obtain a rate of convergence which is valid uniformly for all time our method of proof consists of a scaling argument and elementary energy analysis based on the underlying wave structure 1993 john wiley sons inc

fluid structure partitioned procedures based on robin transmission conditions in this article we design new partitioned procedures for fluid structure interaction problems based on robin type transmission conditions the choice of the coefficient in the robin conditions is justified via simplified models the strategy is effective whenever an incompressible fluid interacts with a relatively thin membrane as in hemodynamics applications we analyze theoretically the new iterative procedures on a model problem which represents a simplified blood vessel system in particular the robin neumann scheme exhibits enhanced convergence properties with respect to the existing partitioned procedures the theoretical results are checked using numerical experimentation

convergence of a fluid structure interaction problem decoupled by a neumann control over a single time step abstract building off of previous analytical results for recasting fluid structure interaction into an optimal control setting an a priori error estimate is given for the optimality system by means of brr theory the convergence of the steepest descent method is proven in a discrete setting for a sufficiently small time step and mesh size a numerical study is included supporting the theoretical rate of convergence over a single time step additional results demonstrate optimal convergence in space and time over several time steps

nonlinear stability of strong planar rarefaction waves for the relaxation approximation of conservation laws in several space dimensions abstract in this paper we show that a strong planar rarefaction wave is nonlinear stable namely it is an attractor for the relaxation approximation of the scalar conservation laws in several space dimensions compared with former results obtained by t p liu 1987 comm math phys 108 153 175 and t luo 1997 j differential equations 133 255 279 our main novelty lies in the fact that the planar rarefaction waves do not need to be small and in the one dimensional case the initial disturbance can also be chosen arbitrarily large

a decoupling algorithm for fluid structure interaction problems based on optimization abstract simulating fluid structure interactions is challenging due to the tight coupling between the fluid and solid substructures explicit and implicit decoupling methods often either fail or require relaxation when densities of the two materials are close in this paper a fluid structure interaction problem is formulated as a least squares problem where the jump in velocities of the two substructures is minimized by a neumann control enforcing the continuity of stress on the interface a decoupling optimization algorithm is discussed which requires few nonlinear solves at each time step and numerical results are presented

decay rate for travelling waves of a relaxation model abstract a relaxation model was proposed in shi jin and zhouping xin comm pure appl math 48 1995 555 563 to approximate the hyperbolic systems numerically under the subcharacteristic condition introduced in t p liu comm math phys 108 1987 153 175 the stability of travelling waves with strong shock profile and integral zero was proved in h l liu j h wang and t yang stability in a relaxation model with nonconvex flux preprint 1996 h l liu and j wang asymptotic stability of travelling wave solutions of a hyperbolic system with relaxation terms preprint 1995 when the original system is scalar in this paper we study the rate of the asymptotic convergence speed of thse travelling wave solutions the analysis applies to the case of a nonconvex flux and when the shock speed coincides with characteristic speed of the state at infinity the decay rate is obtained by applying the energy method and is shown to be the same as the one for the viscous conservation law a matsumura and k nishihara comm math phys 165 1994 83 96

a monolithic mixed finite element method for a fluid structure interaction problem abstract we propose a numerical method for modeling the interaction of a stokes fluid and a linear elastic solid the model problem is expressed in the stress displacement formulation for the linear elastodynamics in the solid region and the stress velocity formulation for the stokes equations in the fluid region these two systems are coupled in such a way that the interface conditions are imposed naturally in the resulting weak formulation which is based on the hellinger reissner variational principle for the time discretization we use a three level scheme for each time step with an exception at the first time step we provide a priori error analysis for fully discrete nonconforming mixed finite element methods and show some numerical results to confirm our theoretical results

an optimization based domain decomposition method for the navier stokes equations an optimization based nonoverlapping domain decomposition method for the solution of the navier stokes equations is presented the crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains the constraints are the navier stokes equations in the subdomains with suitably chosen boundary conditions along the interfaces we show that solutions of the minimization problem exist and derive an optimality system from which these solutions may be determined finite element approximations of the solutions of the optimality system are examined the domain decomposition method is also reformulated as a nonlinear least squares problem and the results of some numerical experiments are given

asymptotic behavior of solutions for p system with relaxation in this paper we consider the asymptotic behavior of solutions for the cauchy problem for p system with relaxationvt ux 0 ut p v x 1e f v u e with initial data v u x 0 v 0 x u 0 x v u v 0 asx i we are interested to show the solutions of e i tend also to the equilibrium rarefaction waves and the traveling waves even if the limits v u of the initial data at x do not satisfy the equilibrium equation i e u f v when the limits of the initial data at infinity satisfy equilibrium states liu 9 studied the stability of rarefaction waves and traveling waves for the general 2 2 hyperbolic conservation laws with relaxation

fully discrete error estimation for a quasi newtonian fluid structure interaction problem we consider an incompressible quasi newtonian fluid structure interaction fsi problem formulated in a monolithic framework where the matching conditions at the moving interface are satisfied the fully discretized fsi system is discussed with detailed analysis for the stability and error estimate as well as numerical experiments that confirm the theoretical results

explicit synchronous partitioned algorithms for interface problems based on lagrange multipliers abstract traditional explicit partitioned schemes exchange boundary conditions between subdomains and can be related to iterative solution methods for the coupled problem as a result these schemes may require multiple subdomain solves acceleration techniques or optimized transmission conditions to achieve sufficient accuracy and or stability we present a new synchronous partitioned method derived from a well posed mixed finite element formulation of the coupled problem we transform the resulting differential algebraic equation dae to a hessenberg index 1 form in which the algebraic equation defines the lagrange multiplier as an implicit function of the states using this fact we eliminate the multiplier and reduce the dae to a system of explicit odes for the states explicit time integration both discretizes this system in time and decouples its equations as a result the temporal accuracy and stability of our formulation are governed solely by the accuracy and stability of the explicit scheme employed and are not subject to additional stability considerations as in traditional partitioned schemes we establish sufficient conditions for the formulation to be well posed and prove that classical mortar finite elements on the interface are a stable choice for the lagrange multiplier we show that in this case the condition number of the schur complement involved in the elimination of the multiplier is bounded by a constant the paper concludes with numerical examples illustrating the approach for two different interface problems

zero relaxation limit to centered rarefaction waves for jin xin relaxation system abstract in this paper we study the zero relaxation limit problem for the following jin xin relaxation system e u t v x 0 v t a 2 u x 1 f u v with initial data i u v x 0 u 0 x v 0 x u v u 0 as x this system was proposed by jin and xin 1995 1 with an interesting numerical origin as the relaxation time tends to zero this system converges to the equilibrium conservation law formally our interest is to study the case where the initial data are allowed to have jump discontinuities such that the corresponding solutions to the equilibrium conservation law contain centered rarefaction waves and the limits u v of the initial data at x do not satisfy the equilibrium equation i e v f u in particular riemann data connected by rarefaction curves are included we show that if the wave strength is sufficiently small then the solution for the relaxation system exists globally in time and converges to the solution of the corresponding rarefaction waves uniformly as the relaxation time goes to zero except for an initial layer

l1nonlinear stability of traveling waves for a hyperbolic system with relaxation where x t r r the unknowns u v belong to r the function f f u is in c r and 0 is a fixed constant relaxation systems often arise in many physical situations let us recall for example gases not in thermodynamic equilibrium vk kinetic theory ce cip pi chromatography raa river flows traffic flows and more general waves wh more recently the study of a special class of hyperbolic systems with relaxation was developed in view of the numerical approximation of discontinuous solutions of conservation laws jx an the 2 2 relaxation hyperbolic systems of conservation laws were first analyzed by liu li2 who justified some nonlinear stability criteria for diffusion waves expansion waves and traveling waves article no 0180

robin based semi implicit coupling in fluid structure interaction stability analysis and numerics in this paper we propose a semi implicit coupling scheme for the numerical simulation of fluid structure interaction systems involving a viscous incompressible fluid the scheme is stable irrespective of the so called added mass effect and allows for conservative time stepping within the structure the efficiency of the scheme is based on the explicit splitting of the viscous effects and geometrical convective nonlinearities through the use of the chorin temam projection scheme within the fluid stability comes from the implicit pressure solid coupling and a specific robin treatment of the explicit viscous solid coupling derived from nitsche s method

fluid dynamic limit for the centered rarefaction wave of the broadwell equation abstract we study the asymptotic equivalence of the 1 d broadwell model of the nonlinear boltzmann equation to its corresponding euler equation of compressible gas dynamics in the limit of small mean free path we consider the case where the initial data are allowed to have jump discontinuities such that the corresponding solutions to the euler equation contain centered rarefaction waves in particular riemann data connected by rarefaction curves are included we show that as long as the initial data are a small perturbation of a non vacuum constant state the solution for the broadwell equation exists globally in time and converges in the small mean free path limit to the solution of the corresponding euler equation uniformly except for an initial layer whose width is essentially the order of the mean free path

convergence to equilibrium for the relaxation approximations of conservation laws we study the cauchy problem for 2 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term special comparison and compactness properties are established by assuming the subcharacteristic condition therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to 0 this research was strongly motivated by the recent numerical investigations of s jin and z xin on the relaxation schemes for conservation laws 1996 john wiley sons inc