characteristic decompositions and interactions of rarefaction waves of 2 d euler equations abstract this paper is concerned with classical solutions to the interaction of two arbitrary planar rarefaction waves for the self similar euler equations in two space dimensions we develop the direct approach started in chen and zheng in press 3 to the problem to recover all the properties of the solutions obtained via the hodograph transformation of li and zheng 2009 14 the direct approach as opposed to the hodograph transformation is straightforward and avoids the common difficulties of the hodograph transformation associated with simple waves and boundaries the approach is made up of various characteristic decompositions of the self similar euler equations for the speed of sound and inclination angles of characteristics

invariant region preserving dg methods for multi dimensional hyperbolic conservation law systems with an application to compressible euler equations abstract an invariant region preserving irp limiter for multi dimensional hyperbolic conservation law systems is introduced as long as the system admits a global invariant region which is a convex set in the phase space it is shown that the order of approximation accuracy is not destroyed by the irp limiter provided the cell average is away from the boundary of the convex set moreover this limiter is explicit and easy for computer implementation a generic algorithm incorporating the irp limiter is presented for high order finite volume type schemes for arbitrarily high order discontinuous galerkin dg schemes to hyperbolic conservation law systems sufficient conditions are obtained for cell averages to remain in the invariant region provided the projected one dimensional system shares the same invariant region as the full multi dimensional hyperbolic system does the general results are then applied to both one and two dimensional compressible euler equations so to obtain high order irp dg schemes numerical experiments are provided to validate the proven properties of the irp limiter and the performance of irp dg schemes for compressible euler equations

semi hyperbolic patches of solutions to the two dimensional euler equations we construct semi hyperbolic patches of solutions in which one family out of two nonlinear families of characteristics starts on sonic curves and ends on transonic shock waves to the two dimensional euler equations this type of solution appears in the transonic flow over an airfoil and guderley reflection and is common in the numerical solutions of riemann problems

a numerical scheme for singular shock solutions and a study of its consistence in the sense of distributions in this paper we present a numerical scheme for the approximation of singular shocks as long as some properties such as boundedness of the velocity are verified when the space step h tends to 0 we prove that the scheme provides approximate solutions that tend to satisfy the equations more precisely when the approximate solutions are plugged into the equations the result tends to 0 when h 0 in the familiar definition of weak solutions with the requirement of a smooth test function these properties can be fully proved for general classes of systems extending the korchinski system that do not have distribution solutions in the usual sense in other cases such as the keyfitz kranzer singular shocks these properties have been checked up to very small values of h these results explain numerical observations on very different systems

a stochastic galerkin method for first order quasilinear hyperbolic systems with uncertainty this paper is concerned with generalized polynomial chaos gpc approximation for first order quasilinear hyperbolic systems with uncertainty the one dimensional 1d hyperbolic system is first symmetrized with the aid of left eigenvector matrix of the jacobian matrix then the gpc stochastic galerkin method is applied to derive a provably symmetrically hyperbolic equations for the gpc expansion coefficients the resulting deterministic gpc galerkin system is discretized by a path conservative finite volume weno scheme in space and a third order total variation diminishing runge kutta method in time the method is further extended to two dimensional 2d quasilinear hyperbolic system with uncertainty where the symmetric hyperbolicity of the one dimensional gpc galerkin system is carried over via an operator splitting technique several numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed gpc stochastic galerkin method

universal approximation using radial basis function networks there have been several recent studies concerning feedforward networks and the problem of approximating arbitrary functionals of a finite number of real variables some of these studies deal with cases in which the hidden layer nonlinearity is not a sigmoid this was motivated by successful applications of feedforward networks with nonsigmoidal hidden layer units this paper reports on a related study of radial basis function rbf networks and it is proved that rbf networks having one hidden layer are capable of universal approximation here the emphasis is on the case of typical rbf networks and the results show that a certain class of rbf networks with the same smoothing factor in each kernel node is broad enough for universal approximation

high resolution finite element analysis of shallow water equations in two dimensions abstract we present in this study the taylor galerkin finite element model to stimulate shallow water equations for bore wave propagation in a domain of two dimensions to provide the necessary precision for the prediction of a sharply varying solution profile the generalized taylor galerkin finite element model is constructed through introduction of four parameters this paper also presents the fundamental theory behind the choice of free parameters one set of parameters is theoretically determined to obtain the high order accurate taylor galerkin finite element model the other set of free parameters is determined using the underlying discrete maximum principle to obtain the low order monotonic taylor galerkin finite element model theoretical study reveals that the higher order scheme exhibits dispersive errors near the discontinuity while lower order scheme dissipates the discontinuity a scheme which has a high resolution shock capturing ability as a built in feature is thus needed in the present study notice that lumping of the mass matrix equations is invoked in the low order scheme to allow simulation of the hydraulic problem with discontinuities we check the prediction accuracy against suitable test problems preferably ones for which exact solutions are available based on numerical results it is concluded that the taylor galerkin flux corrected transport tg fct finite element method can render the technique suitable for solving shallow water equations with sharply varying solution profiles

on the persistency of excitation in radial basis function network identification of nonlinear systems considers radial basis function rbf network approximation of a multivariate nonlinear mapping as a linear parametric regression problem linear recursive identification algorithms applied to this problem are known to converge provided the regressor vector sequence has the persistency of excitation pe property the main contribution of this paper is formulation and proof of pe conditions on the input variables in the rbf network identification the regressor vector is a nonlinear function of these input variables according to the formulated condition the inputs provide pe if they belong to domains around the network node centers for a two input network with gaussian rbf that have typical width and are centered on a regular mesh these domains cover about 25 of the input domain volume the authors further generalize the proposed solution of the standard rbf network identification problem and study affine rbf network identification that is important for affine nonlinear system control for the affine rbf network the author formulates and proves a pe condition on both the system state parameters and control inputs

global solutions of shock reflection by large angle wedges for potential flow when a plane shock hits a wedge head on it experiences a reflectiondiffraction process and then a self similar reflected shock moves outward as the original shock moves forward in time experimental computational and asymptotic analysis has shown that various patterns of shock reflection may occur including regular and mach reflection however most of the fundamental issues for shock reflection have not been understood including the global structure stability and transition of the different patterns of shock reflection therefore it is essential to establish the global existence and structural stability of solutions of shock reflection in order to understand fully the phenomena of shock reflection on the other hand there has been no rigorous mathematical result on the global existence and structural stability of shock reflection including the case of potential flow which is widely used in aerodynamics such problems involve several challenging difficulties in the analysis of nonlinear partial differential equations such as mixed equations of elliptic hyperbolic type free boundary problems and corner singularity where an elliptic degenerate curve meets a free boundary in this paper we develop a rigorous mathematical approach to overcome these difficulties i nvolved and establish a global theory of existence and stability for shock reflection by large angle wedges for potential flow the techniques and ideas developed here will be useful for other nonlinear problems involving similar difficulties

an approach to parametric nonlinear least square optimization and application to task level learning control this paper considers a parametric nonlinear least square nls optimization problem unlike a classical nls problem statement we assume that a nonlinear optimized system depends on two arguments an input vector and a parameter vector the input vector can be modified to optimize the system while the parameter vector changes from one optimization iteration to another and is not controlled the optimization process goal is to find a dependence of the optimal input vector on the parameter vector where the optimal input vector minimizes a quadratic performance index the paper proposes an extension of the levenberg marquardt algorithm for a numerical solution of the formulated problem the proposed algorithm approximates the nonlinear system in a vicinity of the optimum by expanding it into a series of parameter vector functions affine in the input vector in particular a radial basis function network expansion is considered the convergence proof for the algorithm is presented the proposed approach is applied to task level learning control of a two link flexible arm each evaluation of the system in the optimization process means completing a controlled motion of the arm

interaction of rarefaction waves of the two dimensional self similar euler equations we construct classical self similar solutions to the interaction of two arbitrary planar rarefaction waves for the polytropic euler equations in two space dimensions the binary interaction represents a major type of interaction in the two dimensional riemann problems and includes in particular the classical problem of the expansion of a wedge of gas into vacuum based on the hodograph transformation the method employed here involves the phase space analysis of a second order equation and the inversion back to or development onto the physical space

numerical solutions of ideal quantum gas dynamical flows governed by semiclassical ellipsoidal statistical distribution the ideal quantum gas dynamics as manifested by the semiclassical ellipsoidal statistical es equilibrium distribution derived in wu et al wu et al 2012 proc r soc a 468 1799 1823 doi 10 1098 rspa 2011 0673 is numerically studied for particles of three statistics this anisotropic es equilibrium distribution was derived using the maximum entropy principle and conserves the mass momentum and energy but differs from the standard fermi dirac or bose einstein distribution the present numerical method combines the discrete velocity or momentum ordinate method in momentum space and the high resolution shock capturing method in physical space a decoding procedure to obtain the necessary parameters for determining the es distribution is also devised computations of two dimensional riemann problems are presented and various contours of the quantities unique to this es model are illustrated the main flow features such as shock waves expansion waves and slip lines and their complex nonlinear interactions are depicted and found to be consistent with existing calculations for a classical gas

efficient solution algorithms for the riemann problem for real gases a procedure for constructing solutions to the riemann problem for gas dynamics with a general convex equation of state is given approximate procedures involving a local parametrization of the equation of state are introduced in order to calculate numerical fluxes in conservative finite difference schemes this leads to difference schemes which are as accurate and almost as fast as the analogous schemes for polytropic gases numerical results in one and two space variables are presented

regularity of solutions to regular shock reflection for potential flow the shock reflection problem is one of the most important problems in mathematical fluid dynamics since this problem not only arises in many important physical situations but also is fundamental for the mathematical theory of multidimensional conservation laws that is still largely incomplete however most of the fundamental issues for shock reflection have not been understood including the regularity and transition of different patterns of shock reflection configurations therefore it is important to establish the regularity of solutions to shock reflection in order to understand fully the phenomena of shock reflection on the other hand for a regular reflection configuration the potential flow governs the exact behavior of the solution in c 1 1 across the pseudo sonic circle even starting from the full euler flow that is both of the nonlinear systems are actually the same in a physically significant region near the pseudo sonic circle thus it becomes essential to understand the optimal regularity of solutions for the potential flow across the pseudo sonic circle the transonic boundary from the elliptic to hyperbolic region and at the point where the pseudo sonic circle the degenerate elliptic curve meets the reflected shock a free boundary connecting the elliptic to hyperbolic region in this paper we study the regularity of solutions to regular shock reflection for potential flow in particular we prove that the c 1 1 regularity is optimal for the solution across the pseudo sonic circle and at the point where the pseudo sonic circle meets the reflected shock we also obtain the c 2 regularity of the solution up to the pseudo sonic circle in the pseudo subsonic region the problem involves two types of transonic flow one is a continuous transition through the pseudo sonic circle from the pseudo supersonic region to the pseudo subsonic region the other a jump transition through the transonic shock as a free boundary from another pseudo supersonic region to the pseudo subsonic region the techniques and ideas developed in this paper will be useful to other regularity problems for nonlinear degenerate equations involving similar difficulties

comparison of some neural network and scattered data approximations the inverse manipulator kinematics example this paper compares the application of five different methods for the approximation of the inverse kinematics of a manipulator arm from a number of joint angle cartesian coordinate training pairs the first method is a standard feedforward neural network with error backpropagation learning the next two methods are derived from an extended kohonen map algorithm that we combine with shepard interpolation for the forward computation we compare the method of ritter et al for the learning of the extended kohonen map to our own scheme based on gradient descent optimization we also study three scattered data approximation algorithms they include two variants of the radial basis function rbf method hardy s multiquadrics and gaussian rbf we further develop our own local polynomial fit method that could be considered as a modification of mclain s method we propose extensions to the considered scattered data approximation algorithms to make them suitable for vector valued multivariable functions such as the mapping of cartesian coordinates into joint angle coordinates

a direct eulerian grp scheme for relativistic hydrodynamics two dimensional case this paper develops a direct eulerian generalized riemann problem grp scheme for two dimensional 2d relativistic hydrodynamics rhd it is an extension of the grp scheme for one dimensional 1d rhds z c yang p he h z tang j comput phys 230 2011 7964 7987 and the grp scheme for the non relativistic hydrodynamics m ben artzi j q li g warnecke j comput phys 218 2006 19 43 in order to derive the direct eulerian grp scheme the local grp of the split 2d rhd equations in the eulerian formulation has to be directly resolved by using corresponding riemann invariants and rankine hugoniot jump conditions so that the crucial and delicate lagrangian treatment in the original grp scheme m ben artzi j falcovitz j comput phys 55 1984 1 32 may be avoided an important difference of resolving the grp of the split 2d rhd equations from the grp of the 1d rhd equations or the non relativistic hydrodynamical equations is coming from the fact that the flow regions across the shock or rarefaction wave in the grp of the split 2d rhd equations are nonlinearly coupled through the lorentz factor which is also built in terms of the tangential velocities it is a purely multi dimensional relativistic feature several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed 2d grp scheme

the structure of solutions near a sonic line in gas dynamics via the pressure gradient equation abstract a key issue in gas dynamics in two space dimensions is the regularity of solutions near a sonic curve we build a large class of regular solutions with given boundary conditions on the sonic line the modeling equation is the pressure gradient equation which is the same as euler system when the parameters of the gas are pushed to certain extreme we use a novel set of coordinates involving both the space time and state variables to split regular terms from singular terms in the analysis

on the two dimensional gas expansion for compressible euler equations we investigate the problem of two dimensional unsteady expansion of an inviscid polytropic gas which can be interpreted as the collapse of a wedge shaped dam containing water initially with a uniform velocity we model this problem by isentropic euler equations the flow is quasi stationary and using hodograph transform we describe it by a partial differential equation of second order in the state space if it is irrotational initially furthermore this equation is reduced to a linearly degenerate system of three partial differential equations with inhomogeneous source terms these properties are used to prove that the flow is globally smooth when a wedge of gas expands into a vacuum and to analyze that shocks may appear in the interaction of four planar rarefaction waves

improving the generalization properties of radial basis function neural networks an important feature of radial basis function neural networks is the existence of a fast linear learning algorithm in a network capable of representing complex nonlinear mappings satisfactory generalization in these networks requires that the network mapping be sufficiently smooth we show that a modification to the error functional allows smoothing to be introduced explicitly without significantly affecting the speed of training a simple example is used to demonstrate the resulting improvement in the generalization properties of the network

transonic shock formation in a rarefaction riemann problem for the 2d compressible euler equations it is perhaps surprising for a shock wave to exist in the solution of a rarefaction riemann problem for the compressible euler equations in two space dimensions we present numerical evidence and generalized characteristic analysis to establish the existence of a shock wave in such a 2d riemann problem defined by the interaction of four rarefaction waves we consider both the customary configuration of waves at the right angle and also an oblique configuration for the rarefaction waves two distinct mechanisms for the formation of a shock wave are discovered as the angle between the waves is varied

on coupled wave equation of kirchhoff type with nonlinear boundary damping and memory term in this paper we prove the existence of the solution to the mixed problem for coupled wave equation of kirchhoff type with nonlinear boundary damping and memory term moreover we discuss the uniform decay of the solution

a practical bayesian framework for backpropagation networks a quantitative and practical bayesian framework is described for learning of mappings in feedforward networks the framework makes possible 1 objective comparisons between solutions using alternative network architectures 2 objective stopping rules for network pruning or growing procedures 3 objective choice of magnitude and type of weight decay terms or additive regularizers for penalizing large weights etc 4 a measure of the effective number of well determined parameters in a model 5 quantified estimates of the error bars on network parameters and on network output and 6 objective comparisons with alternative learning and interpolation models such as splines and radial basis functions the bayesian evidence automatically embodies occam s razor penalizing overflexible and overcomplex models the bayesian approach helps detect poor underlying assumptions in learning models for learning models well matched to a problem a good correlation between generalization ability and the bayesian evidence is obtained

some new finite difference inequalities abstract in this paper we establish some new finite difference inequalities related to a certain integral inequality arising in the theory of differential equations the inequalities obtained here can be used as handy tools in the theory of some new classes of finite difference and sum difference equations

global solution of an initial value problem for two dimensional compressible euler equations abstract this paper is concerned with the existence of global continuous solutions of the expansion of a wedge of gas into a vacuum for compressible euler equations by hodograph transformation we first prove that the flow is governed by a partial differential equation of second order which is further reduced to a system of two nonhomogeneous linearly degenerate equations in the phase space under an irrotationality condition then this conclusion is applied to solving the problem that a wedge of gas expands into a vacuum which is actually a goursat type problem for these two equations in the supersonic domain

variants of relaxed schemes and two dimensional gas dynamics we present a relaxed scheme with more precise information about local speeds of propagation and a multidimensional construction of the cell averages the physical domain of dependence is simulated correctly and high resolution is maintained relaxation schemes have advantages that include high resolution simplicity and explicitly no approximate riemann solvers and no characteristic decomposition is necessary performance of the scheme is illustrated by tests on two dimensional euler equations of gas dynamics

solution of two dimensional riemann problems of gas dynamics by positive schemes the positivity principle and positive schemes to solve multidimensional hyperbolic systems of conservation laws have been introduced in x d liu and p d lax j fluid dynam 5 1996 pp 133 156 some numerical experiments presented there show how well the method works in this paper we use positive schemes to solve riemann problems for two dimensional gas dynamics

on the gibbs phenomenon and its resolution the nonuniform convergence of the fourier series for discontinuous functions and in particular the oscillatory behavior of the finite sum was already analyzed by wilbraham in 1848 this was later named the gibbs phenomenon this article is a review of the gibbs phenomenon from a different perspective the gibbs phenomenon as we view it deals with the issue of recovering point values of a function from its expansion coefficients alternatively it can be viewed as the possibility of the recovery of local information from global information the main theme here is not the structure of the gibbs oscillations but the understanding and resolution of the phenomenon in a general setting the purpose of this article is to review the gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function with the same order of accuracy as in the smooth case this is done by using the finite expansion series to construct a different rapidly convergent approximation

a direct eulerian grp scheme for compressible fluid flows a direct eulerian generalized riemann problem grp scheme is derived for compressible fluid flows riemann invariants are introduced as the main ingredient to resolve the generalized riemann problem grp directly for the eulerian formulation the crucial auxiliary lagrangian scheme in the original grp scheme is not necessary in the present framework the delicate sonic cases can be easily treated and the extension to multidimensional cases is obtained using the dimensional splitting technique

remapping free ale type kinetic method for flow computations based on the integral form of the fluid dynamic equations a finite volume kinetic scheme with arbitrary control volume and mesh velocity is developed different from the earlier unified moving mesh gas kinetic method c q jin k xu an unified moving grid gas kinetic method in eulerian space for viscous flow computation j comput phys 222 2007 155 175 the coupling of the fluid equations and geometrical conservation laws has been removed in order to make the scheme applicable for any quadrilateral or unstructured mesh rather than parallelogram in 2d case since a purely lagrangian method is always associated with mesh entangling in order to avoid computational collapsing in multidimensional flow simulation the mesh velocity is constructed by considering both fluid velocity lagrangian methodology and diffusive velocity regenerating eulerian mesh function therefore we obtain a generalized arbitrary lagrangian eulerian ale method by properly designing a mesh velocity instead of re generating a new mesh after distortion as a result the remapping step to interpolate flow variables from old mesh to new mesh is avoided the current method provides a general framework which can be considered as a remapping free ale type method since there is great freedom in choosing mesh velocity in order to improve the accuracy and robustness of the method the adaptive moving mesh method h z tang t tang adaptive mesh methods for one and two dimensional hyperbolic conservation laws siam j numer anal 41 2003 487 515 can be also used to construct a mesh velocity to concentrate mesh to regions with high flow gradients

on uniform decay of solutions for wave equation of kirchhoff type with nonlinear boundary damping and memory source term in this paper we prove the existence of solution to the mixed problem for wave equation of kirchhoff type with nonlinear boundary damping and memory term moreover we discuss the uniform decay of solution

predicting the future advantages of semilocal units in investigating gaussian radial basis function rbf networks for their ability to model nonlinear time series we have found that while rbf networks are much faster than standard sigmoid unit backpropagation for low dimensional problems their advantages diminish in high dimensional input spaces this is particularly troublesome if the input space contains irrelevant variables we suggest that this limitation is due to the localized nature of rbfs to gain the advantages of the highly nonlocal sigmoids and the speed advantages of rbfs we propose a particular class of semilocal activation functions that is a natural interpolation between these two families we present evidence that networks using these gaussian bar units avoid the slow learning problem of sigmoid unit networks and very importantly are more accurate than rbf networks in the presence of irrelevant inputs on the mackey glass and coupled lattice map problems the speedup over sigmoid networks is so dramatic that the difference in training time between rbf and gaussian bar networks is minor gaussian bar architectures that superpose composed gaussians gaussians of gaussians to approximate the unknown function have the best performance we postulate that an interesing behavior displayed by gaussian bar functions under gradient descent dynamics which we call automatic connection pruning is an important factor in the success of this representation

on discretely entropy conservative and entropy stable discontinuous galerkin methods abstract high order methods based on diagonal norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic pdes 1 2 these methods can also be interpreted as nodal discontinuous galerkin methods with diagonal mass matrices 3 4 5 6 in this work we describe how use flux differencing quadrature based projections and sbp like operators to construct discretely entropy conservative schemes for dg methods under more arbitrary choices of volume and surface quadrature rules the resulting methods are semi discretely entropy conservative or entropy stable with respect to the volume quadrature rule used numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible euler equations in one and two dimensions

yosida approximation and nonlinear hyperbolic equation we are concerned with the initial value problem for nonlinear evolution equations of the form u t m a 1 2 u t 2 au t u t f t on 0 u 0 u o u o u 1 here a is a nonnegative selfadjoint operator in a real hilbert space h 0 is a constant and m is a c 1 class function satisfying m r m o 0 for r 0 with m o constant the equation has its motivation in the mathematical description of small amplitude vibrations of an elastic string

comparison of the generalized riemann solver and the gas kinetic scheme for inviscid compressible flow simulations the generalized riemann problem grp scheme for the euler equations and gas kinetic scheme gks for the boltzmann equation are two high resolution shock capturing schemes for fluid simulations the difference is that one is based on the characteristics of the inviscid euler equations and their wave interactions and the other is based on the particle transport and collisions the similarity between them is that both methods can use identical muscl type initial reconstructions around a cell interface and the spatial slopes on both sides of a cell interface involve in the gas evolution process and the construction of a time dependent flux function although both methods have been applied successfully to the inviscid compressible flow computations their performances have never been compared since both methods use the same initial reconstruction any difference is solely coming from different underlying mechanism in their flux evaluation therefore such a comparison is important to help us to understand the correspondence between physical modeling and numerical performances since grp is so faithfully solving the inviscid euler equations the comparison can be also used to show the validity of solving the euler equations itself the numerical comparison shows that the grp exhibits a slightly better computational efficiency and has comparable accuracy with gks for the euler solutions in 1d case but the gks is more robust than grp for the 2d high mach number flow simulations the gks is absent from the shock instability and converges to the steady state solutions faster than the grp the grp has carbuncle phenomena likes a cloud hanging over exact riemann solvers the grp and gks use different physical processes to describe the flow motion starting from a discontinuity one is based on the assumption of equilibrium state with infinite number of particle collisions and the other starts from the non equilibrium free transport process to evolve into an equilibrium one through particle collisions the different mechanism in the flux evaluation deviates their numerical performance through this study we may conclude scientifically that it may not be valid to use the euler equations as governing equations to construct numerical fluxes in a discretized space with limited cell resolution to adapt the navier stokes ns equations is not valid either because the ns equations describe the flow behavior on the hydrodynamic scale and have no any corresponding physics starting from a discontinuity this fact alludes to the consistency of the euler and navier stokes equations with the continuum assumption and the necessity of a direct modeling of the physical process in the discretized space in the construction of numerical scheme when modeling very high mach number flows the development of numerical algorithm is similar to the modeling process in deriving the governing equations but the control volume here cannot be shrunk to zero

on the divergence free condition in godunov type schemes for ideal magnetohydrodynamics the upwind constrained transport method we present a general framework to design godunov type schemes for multidimensional ideal magnetohydrodynamic mhd systems having the divergence free relation and the related properties of the magnetic field b as built in conditions our approach mostly relies on the constrained transport ct discretization technique for the magnetic field components originally developed for the linear induction equation which assures b num 0 and its preservation in time to within machine accuracy in a finite volume setting we show that the ct formalism when fully exploited can be used as a general guideline to design the reconstruction procedures of the b vector field to adapt standard upwind procedures for the momentum and energy equations avoiding the onset of numerical monopoles of o 1 size and to formulate approximate riemann solvers for the induction equation this general framework will be named here upwind constrained transport uct to demonstrate the versatility of our method we apply it to a variety of schemes which are finally validated numerically and compared a novel implementation for the mhd case of the second order roetype positive scheme by lin and lax j comput fluid dyn 5 1996 133 and both the second and third order versions of a central type mhd scheme presented by londrillo and del zanna astrophys j 530 2000 508 where the basic uct strategies have been first outlined

smoothing effect and propagations of singularities for viscoelastic plates abstract we study dissipative models for plates and we show that the solutions have a smoothing effect on the initial data for viscous plates while for materials with memory the solution propagates singularities that is the solution of the plate equation of memory type is as regular as the initial data moreover we show that when both dissipations are present the memory type prevails in the sense that the solution propagates singularities finally we prove the existence of global solutions for non linear dissipative equations with small data which decay exponentially as time goes to infinity

characteristic decouplings and interactions of rarefaction waves of 2d euler equations abstract this paper is concerned with classical solutions for the interaction of two arbitrary planar rarefaction waves for the self similar euler equations in two space dimensions the approach is made up of various characteristic decompositions of the self similar euler equations for the speed of sound and inclination angles of characteristics reported in an earlier paper j li z yang y zheng characteristic decompositions and interactions of rarefaction waves of 2 d euler equations j differential equations 250 2011 782 798 here we report a new decomposition of the characteristics that decouples the two derivatives of the speed of sound along the two families of characteristics and we apply the decomposition to refine the global existence theory and convexity properties of the characteristics

weak asymptotic methods for scalar equations and systems abstract in this paper we show how one can construct families of continuous functions which satisfy asymptotically scalar equations with discontinuous nonlinearity and systems having irregular solutions this construction produces weak asymptotic methods which are issued from maslov asymptotic analysis we obtain a sequence of functions which tend to satisfy the equation s in the weak sense in the space variable and in the strong sense in the time variable to this end we reduce the partial differential equations to a family of ordinary differential equations in a classical banach space for scalar equations we prove that the initial value problem is well posed in the l 1 sense for the approximate solutions we construct then we prove that this method gives back the widely accepted solutions when they are known for systems we obtain existence in the general case and uniqueness in the analytic case using an abstract cauchy kovalevska theorem

an efficient direct solver for rarefied gas flows with arbitrary statistics a new numerical methodology associated with a unified treatment is presented to solve the boltzmann bgk equation of gas dynamics for the classical and quantum gases described by the bose einstein and fermi dirac statistics utilizing a class of globally stiffly accurate implicit explicit runge kutta scheme for the temporal evolution associated with the discrete ordinate method for the quadratures in the momentum space and the weighted essentially non oscillatory method for the spatial discretization the proposed scheme is asymptotic preserving and imposes no non linear solver or requires the knowledge of fugacity and temperature to capture the flow structures in the hydrodynamic euler limit the proposed treatment overcomes the limitations found in the work by yang and muljadi 2011 33 due to the non linear nature of quantum relations and can be applied in studying the dynamics of a gas with internal degrees of freedom with correct values of the ratio of specific heat for the flow regimes for all knudsen numbers and energy wave lengths the present methodology is numerically validated with the unified treatment by the one dimensional shock tube problem and the two dimensional riemann problems for gases of arbitrary statistics descriptions of ideal quantum gases including rotational degrees of freedom have been successfully achieved under the proposed methodology

nonlinear generalized functions and jump conditions for a standard one pressure liquid gas model abstract the mixture of a liquid and a gas is classically represented by one pressure models these models are a system of pdes in nonconservative form and shock wave solutions do not make sense within the theory of distributions they give rise to products of distributions that are not defined within distribution theory but they make sense by applying a theory of nonlinear generalized functions to these equations in contrast to the familiar case of conservative systems the jump conditions cannot be calculated a priori jump conditions for these nonconservative systems can be obtained using the theory of nonlinear generalized functions by inserting some adequate physical information into the equations the physical information that we propose to insert for the one pressure models of a mixture of a liquid and a gas is a natural mathematical expression in the theory of nonlinear generalized functions of the fact that liquids are practically incompressible while gases are very compressible and so they do not satisfy equally well their respective state laws on the shock waves this modelization gives well defined explicit jump conditions the great numerical difficulty for solving numerically nonconservative systems is due to the fact that slightly different numerical schemes can give significantly different results the jump conditions obtained above permit to select the numerical schemes and validate those that give numerical solutions that satisfy these jump conditions which can be an important piece of information in the absence of other explicit discontinuous solutions and of precise observational results we expose with care the mathematical originality of the theory of nonlinear generalized functions an original abstract analysis issued by the leopoldo nachbin team on infinite dimensional holomorphy that permits to state mathematically physical facts that cannot be formulated within distribution theory and are the key for the removal of ambiguities that classically appear when one tries to calculate on multiplications of distributions that occur in the differential equations of physics

limits of nonlinear weak asymptotic methods we propose a mathematical limit of l1 stable weak asymptotic methods a family of l1 stable approximate solutions is transformed into a normal family of holomorphic functions defined in a complex domain having the real space on its boundary this provides a holomorphic function which is the same mathematical object as the solutions from explicit calculations the weak limit of the approximate solutions from weak asymptotic methods in the space of bounded radon measures is recovered as a boundary value of this holomorphic function

conservative form of interpolated differential operator scheme for compressible and incompressible fluid dynamics the proposed scheme which is a conservative form of the interpolated differential operator scheme ido cf can provide high accurate solutions for both compressible and incompressible fluid equations spatial discretizations with fourth order accuracy are derived from interpolation functions locally constructed by both cell integrated values and point values these values are coupled and time integrated by solving fluid equations in the flux forms for the cell integrated values and in the derivative forms for the point values the ido cf scheme exactly conserves mass momentum and energy retaining the high resolution more than the non conservative form of the ido scheme a direct numerical simulation of turbulence is carried out with comparable accuracy to that of spectral methods benchmark tests of riemann problems and lid driven cavity flows show that the ido cf scheme is immensely promising in compressible and incompressible fluid dynamics studies

gaussian networks for direct adaptive control a direct adaptive tracking control architecture is proposed and evaluated for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible the architecture uses a network of gaussian radial basis functions to adaptively compensate for the plant nonlinearities under mild assumptions about the degree of smoothness exhibit by the nonlinear functions the algorithm is proven to be globally stable with tracking errors converging to a neighborhood of zero a constructive procedure is detailed which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy a stable weight adjustment mechanism is determined using lyapunov theory the network construction and performance of the resulting controller are illustrated through simulations with example systems

a third order accurate direct eulerian grp scheme for the euler equations in gas dynamics the paper proposes and implements a third order accurate direct eulerian generalized riemann problem grp scheme for one and two dimensional 1d 2d euler equations in gas dynamics it is an extension of the second order accurate grp scheme proposed in ben artzi et al 2006 5 the approximate states in numerical fluxes of the third order accurate grp scheme are derived by using the higher order weno reconstruction of the initial data the limiting values of the time derivatives of the solutions at the singularity point and the jacobian matrix besides the limiting values of the first order time derivatives of fluid variables the second order time derivatives are also needed in developing the present grp scheme and obtained by directly and analytically resolving the local grp in the eulerian formulation via two main ingredients i e the riemann invariants and rankine hugoniot jump conditions unfortunately for the sonic case that the transonic rarefaction wave appears in the grp the jacobian matrix is singular on the sonic line to this end those approximate states are given in a different way that is based on the analytical resolution of the transonic rarefaction wave and the local quadratic polynomial interpolation the 2d grp scheme is implemented by using the third order accurate time splitting method several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed grp scheme in comparison to the second order accurate grp scheme

linear schrodinger evolution equations with moving coulomb singularities abstract the cauchy problem for schrodinger evolution equations with a finite number of moving coulomb singularities is investigated the case of a single singularity has been studied by several authors see e g baudouin et al 2005 1 and okazawa et al 2010 19 however it seems to be no previous work on plural singularities we shall show that the problem has a unique classical solution by using a time dependent linear transformation of the unknown function which locally freezes the motion of the whole singularities under the simplest collisionless condition in fact a new existence and uniqueness theorem is available for the transformed problem such an abstract framework is established from the viewpoint of linear evolution equations of hyperbolic type in a hilbert space as an innovative modification of those in okazawa 1998 17 and okazawa and yoshii 2011 20

pyclaw accessible extensible scalable tools for wave propagation problems development of scientific software involves tradeoffs between ease of use generality and performance we describe the design of a general hyperbolic pde solver that can be operated with the convenience of matlab yet achieves efficiency near that of hand coded fortran and scales to the largest supercomputers this is achieved by using python for most of the code while employing automatically wrapped fortran kernels for computationally intensive routines and using python bindings to interface with a parallel computing library and other numerical packages the software described here is pyclaw a python based structured grid solver for general systems of hyperbolic pdes k t mandli et al pyclaw software version 1 0 http numerics kaust edu sa pyclaw 2011 pyclaw provides a powerful and intuitive interface to the algorithms of the existing fortran codes clawpack and sharpclaw simplifying code development and use while providing massive parallelism and scalable solvers via the petsc library the

asmo dan algorithm for adaptive spline modelling of observation data nonlinear system identification by modelling the underlying relationships in observation data is an important application area for artificial neural networks and other learning paradigms splines have been used for scattered data interpolation but the applications have mainly been restricted to low dimensional input spaces this paper describes asmod a new learning paradigm for higher dimensional data 3 based on b spline interpolation the models can be trained online and a method for step wise model refinement is applied during model training for gradually increasing the modelling capability until the desired or best possible accuracy is obtained for every refinement step a number of possible refinement actions are evaluated and the one that gives the highest improvement of the model accuracy is chosen the model structure is hence adapted to the modelling problem giving a model of small size and high accuracy asmod has very efficient implementations on serial computers the scheme has been

a simple numerical scheme for the 3 d system of ideal gases and a study of approximation in the sense of distributions we present a 3 d numerical scheme for the approximation of the system of gas dynamics we prove that as long as the boundedness of the velocity field in the cfl condition and the positiveness of the total energy are numerically verified when the space step tends to 0 the scheme provides a numerical solution which satisfies the conservation laws in the sense of distributions modulo a small remainder of order one in the space step and a slightly weaker order for the state law in the case of irregular approximate solutions in the case of classical tests sod woodward colella toro one provides numerical verifications that these conditions are verified even for very small values of the space step and correspondingly one observes that the scheme produces the known exact solutions with great precision full proofs are given in the pressureless case the scheme produces the numerical results on the six 2 d riemann problems presented by p d lax p d lax mathematics and physics bull ams 45 1 2008 135 152 up to the smallest detail therefore this result contributes to a mathematical justification of the fact that the numerical results presented by p d lax on these 2 d riemann problems are approximations of a solution this simple order one low cost 3 d scheme is obtained from the convection pressure correction method proposed by le roux et al r baraille g bourdin f dubois a y le roux une version a pas fractionnaire du schema de godunov pour l hydrodynamique comptes rendus acad sci paris 314 1992 147 152

modeled boltzmann equation and its application to shock capturing simulation a modified equilibrium distribution function for the bhatnagar gross krook type modeled boltzmann equation has recently been proposed the function was deduced using acoustics scaling to normalize the equation and allowed a correct recovery of similarly normalized euler equations it is a combination of a maxwellian distribution plus three other terms that are moments of particle velocity the lattice counterpart of the modified equilibrium distribution function also led to an exact recovery of the euler equations therefore there is no mach number limitation in the entire approach this lattice counterpart was able to replicate aeroacoustics problems involving vorticity acoustic and entropy acoustic interactions correctly and the simulations were carried out using a finite difference lattice boltzmann method employing only a two dimensional nine velocity lattice thus formulated the numerical scheme has no arbitrary constants and all calculations were carried out using one single relaxation time and a set of constants derived from the analysis this paper investigates the validity and extent of the formulation to capture shocks and resolve contact discontinuity and expansion waves in one and two dimensional riemann problems the simulations are carried out using the same two dimensional nine velocity lattice and identical set of constants and relaxation time they are compared with theoretical results and those obtained by solving the euler equations directly using harten s first order numerical scheme good agreement is obtained for all test cases however the modified equilibrium distribution function is not suitable for shock structure simulation for that an exact recovery of the navier stokes equations is required

a multiresolution finite volume scheme for two dimensional hyperbolic conservation laws in this paper high resolution finite volume schemes are combined with an adaptive mesh technique inspired by multiresolution analysis to improve the computational efficiency for two dimensional hyperbolic conservation laws the method is conservative moreover it is stable which is proven numerically in this paper the computational grid is dynamically adapted so that higher spatial resolution is automatically allocated to regions where strong gradients are observed using this proposed scheme we compute several two dimensional model problems and a compressive rate ranging from about 5 10 is observed in all simulations

asymptotic study of the initial value problem to a standard one pressure model of multifluid flows in nondivergence form abstract we construct families of approximate solutions to the initial value problem and provide complete mathematical proofs that they tend to satisfy the standard system of isothermal one pressure two fluid flows in 1 d when the data are l 1 in densities and l in velocities to this end we use a method that reduces this system of pdes to a family of systems of four odes in banach spaces whose smooth solutions are these approximate solutions this method is constructive using standard numerical methods for odes one can easily and accurately compute these approximate solutions which therefore from the mathematical proof can serve for comparison with numerical schemes one observes agreement with previously known solutions from scientific computing evje and flatten 2003 16 we show that one recovers the solutions of these authors exactly in one case with a slight difference in another case then we propose an efficient numerical scheme for the original system of two fluid flows and show it gives back exactly the same results as the theoretical solutions obtained above

a method of projection of delta waves in a godunov scheme and application to pressureless fluid dynamics some systems of pdes such as the one of pressureless fluid dynamics show delta waves in the solution of the riemann problem a method of projection of these delta waves in a godunov scheme is proposed it provides a modification of the original le roux et al scheme in case of changes in sign of velocity r baraille g bourdin f dubois and a y le roux c r acad sci paris 314 1992 pp 147 152 stability and convergence of the scheme are proved in one space dimension for the system of pressureless fluids as an application this method has been extended to classical systems of fluid dynamics used for the numerical simulation of large structure formation in cosmology in presence of expansion of the background this method of projection of delta waves can also be applied to systems of conservation laws that have delta waves in the solution to the riemann problem