fast high order ader schemes for linear hyperbolic equations a reformulation of the ader approach arbitrary high order schemes using derivatives for linear hyperbolic pde s is presented this reformulation leads to a drastic decrease of the computational effort a formula for the construction of ader schemes that are arbitrary high order accurate in space and time is given the accuracy for some selected schemes is shown numerically for the two dimensional linearized euler equations as a mathematical model for noise propagation in the time domain in aeroacoustics

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

a time domain numerical modeling of two dimensional wave propagation in porous media with frequency dependent dynamic permeability an explicit finite difference scheme is presented for solving the two dimensional biot equations of poroelasticity across the full range of frequencies the key difficulty is to discretize the johnson koplik dashen jkd model which describes the viscous dissipations in the pores indeed the time domain version of biot jkd model involves order 1 2 fractional derivatives which amount to a time convolution product to avoid storing the past values of the solution a diffusive representation of fractional derivatives is used the convolution kernel is replaced by a finite number of memory variables that satisfy local in time ordinary differential equations the coefficients of the diffusive representation follow from an optimization procedure of the dispersion relation then various methods of scientific computing are applied the propagative part of the equations is discretized using a fourth order finite difference scheme whereas the diffusive part is solved exactly an immersed interface method is implemented to discretize the geometry on a cartesian grid and also to discretize the jump conditions at interfaces numerical experiments are proposed in various realistic configurations

a second order godunov type scheme for compressible fluid dynamics abstract a second order accurate scheme for the integration in time of the conservation laws of compressible fluid dynamics is presented two related versions are proposed one lagrangian and the second direct eulerian they both share the common ingredient which is a full analytic solution for the time derivatives of flow quantities at a jump discontinuity assuming initial nonvanishing slopes on both sides while this solution is an extension of the solution to the classical riemann problem the resulting schemes are second order extensions of godunov s methods in both cases they are very simple to implement in computer codes several numerical examples are shown where the only additional mechanism is a simple monotonicity algorithm

brief paper stochastic h2 h control for discrete time systems with state and disturbance dependent noise this paper is to study the discrete time stochastic h 2 h control with state and external disturbance dependent noise a necessary and sufficient condition for the existence of h 2 h control is presented which transforms the h 2 h controller design into solving four coupled matrix valued equations moreover a recursive algorithm for solving the four coupled matrix valued equations is also given

time domain numerical modeling of wave propagation in 2d heterogeneous porous media this paper deals with the numerical modeling of wave propagation in porous media described by biot s theory the viscous efforts between the fluid and the elastic skeleton are assumed to be a linear function of the relative velocity which is valid in the low frequency range the coexistence of propagating fast compressional wave and shear wave and of a diffusive slow compressional wave makes numerical modeling tricky to avoid restrictions on the time step the biot s system is splitted into two parts the propagative part is discretized by a fourth order ader scheme while the diffusive part is solved analytically near the material interfaces a space time mesh refinement is implemented to capture the small spatial scales related to the slow compressional wave the jump conditions along the interfaces are discretized by an immersed interface method numerical experiments and comparisons with exact solutions confirm the accuracy of the numerical modeling the efficiency of the approach is illustrated by simulations of multiple scattering

interval estimates for the ratio of the means of two normal populations with variances related to the means summary a procedure is given for estimating the ratio of the means of two populations using the data from two independent random samples when the observations are normally distributed with population variances that are related to the population means

robust h control for a class of nonlinear discrete time delay stochastic systems with missing measurements this paper is concerned with the problem of robust h output feedback control for a class of uncertain discrete time delayed nonlinear stochastic systems with missing measurements the parameter uncertainties enter into all the system matrices the time varying delay is unknown with given low and upper bounds the nonlinearities satisfy the sector conditions and the missing measurements are described by a binary switching sequence that obeys a conditional probability distribution the problem addressed is the design of an output feedback controller such that for all admissible uncertainties the resulting closed loop system is exponentially stable in the mean square for the zero disturbance input and also achieves a prescribed h performance level by using the lyapunov method and stochastic analysis techniques sufficient conditions are first derived to guarantee the existence of the desired controllers and then the controller parameters are characterized in terms of linear matrix inequalities lmis a numerical example is exploited to show the usefulness of the results obtained

higher order multilevel framework for ader scheme in computational aeroacoustics abstract the versatile arbitrary derivative ader scheme is cast in a multilevel framework ml ader for fast solution of system of linear hyperbolic partial differential equations the solution is cycled through spatial operators of varying accuracy while retaining highest order accuracy by the use of a forcing function accuracy analysis of the multilevel framework including in the ml ader form is carried out in time domain as well as frequency domain results are obtained for benchmark problems in computational aeroacoustics at a much reduced computational cost

junction generalized riemann problem for stiff hyperbolic balance laws in networks in this paper we design a new implicit solver for the junction generalized riemann problem j grp which is based on a recently proposed implicit method for solving the generalized riemann problem grp for systems of hyperbolic balance laws we use the new j grp solver to construct an ader scheme that is globally explicit locally implicit and with no theoretical accuracy barrier in both space and time the resulting ader scheme is able to deal with stiff source terms and can be applied to non linear systems of hyperbolic balance laws in domains consisting on networks of one dimensional sub domains in this paper we specifically apply the numerical techniques to networks of blood vessels we report on a test problem with exact solution for a simplified network of three vessels meeting at a single junction which is then used to carry out a systematic convergence rate study of the proposed high order numerical methods schemes up to fifth order of accuracy in space and time are implemented and tested we then show the ability of the ader scheme to deal with stiff sources through a numerical simulation in a network of vessels an application to a physical test problem consisting of a network of 37 compliant silicon tubes arteries and 21 junctions reveals that it is imperative to use high order methods at junctions in order to preserve the desired high order of accuracy in the full computational domain for example it is demonstrated that a second order method throughout gives comparable results to a method that is fourth order in the interior of the domain and first order at junctions we are concerned with stiff hyperbolic balance laws in networks an implicit solver for the junction generalized riemann problem is proposed ader schemes of arbitrary accuracy in space and time for networks are constructed convergence rates studies of schemes up to fifth order are carried out it is necessary to match the accuracy at junctions to that of the rest of the domain

heterogeneous domain decomposition for computational aeroacoustics this paper presents a strategy to accelerate the direct simulation of aeroacoustic problems in terms of cpu time the key idea is to introduce a heterogeneous domain decomposition the whole computational domain is subdivided into smaller domains in each of these subdomains the equations the discretization the mesh and the time step may be different and are adapted to the local behavior of the solution to reduce the total number of elements we propose the use of high order methods here the class of arbitrary high order using derivatives finite volume schemes on structured meshes and arbitrary high order using derivatives discontinuous galerkin methods on unstructured meshes seem a good choice to us the coupling procedure is validated and numerical results for the interface transmission problem and the single airfoil gust response problem from 4th computational aeroacoustics workshop on benchmark problems cp 2004 212954 nasa 2004 are presented together with the acoustic scattering problem at a cylinder and at multiple objects the coupling approach proves to be especially efficient for the propagation of sound in large domains

stochastic h sub 2 h sub spl infin control with state dependent noise this paper discusses the stochastic h sub 2 h sub spl infin control problem with state dependent noise by means of the stabilization exact observability and stochastic detectability of stochastic systems the infinite horizon stochastic h sub 2 h sub spl infin control design is developed for the finite horizon h sub 2 h sub spl infin control problem our results generalize the corresponding deterministic ones to the stochastic models finally the observer based suboptimal stochastic h sub 2 h sub spl infin control is discussed in which the state variables cannot be measured directly and a feasible design algorithm is proposed

mixed h sub 2 h sub spl infin fuzzy output feedback control design for nonlinear dynamic systems an lmi approach this study introduces a mixed h sub 2 h sub spl infin fuzzy output feedback control design method for nonlinear systems with guaranteed control performance first the takagi sugeno fuzzy model is employed to approximate a nonlinear system next based on the fuzzy model a fuzzy observer based mixed h sub 2 h sub spl infin controller is developed to achieve the suboptimal h sub 2 control performance with a desired h sub spl infin disturbance rejection constraint a robust stabilization technique is also proposed to override the effect of approximation error in the fuzzy approximation procedure by the proposed decoupling technique and two stage procedure the outcome of the fuzzy observer based mixed h sub 2 h sub spl infin control problem is parametrized in terms of the two eigenvalue problems evps one for observer and the other for controller the evps can be solved very efficiently using the linear matrix inequality lmi optimization techniques a simulation example is given to illustrate the design procedures and performances of the proposed method

numerical modeling of 1d transient poroelastic waves in the low frequency range propagation of transient mechanical waves in porous media is numerically investigated in 1d the framework is the linear biot model with frequency independent coefficients the coexistence of a propagating fast wave and a diffusive slow wave makes numerical modeling tricky a method combining three numerical tools is proposed a fourth order ader scheme with time splitting to deal with the time marching a space time mesh refinement to account for the small scale evolution of the slow wave and an interface method to enforce the jump conditions at interfaces comparisons with analytical solutions confirm the validity of this approach

derivative riemann solvers for systems of conservation laws and ader methods in this paper we first briefly review the semi analytical method e f toro v a titarev solution of the generalized riemann problem for advection reaction equations proc roy soc london 458 2018 2002 271 281 for solving the derivative riemann problem for systems of hyperbolic conservation laws with source terms next we generalize it to hyperbolic systems for which the riemann problem solution is not available as an application example we implement the new derivative riemann solver in the high order finite volume ader advection schemes we provide numerical examples for the compressible euler equations in two space dimensions which illustrate robustness and high accuracy of the resulting schemes

high order cut discontinuous galerkin methods with local time stepping for acoustics we propose a method to solve the acoustic wave equation on an immersed domain using the hybridizable discontinuous galerkin method for spatial discretization and the arbitrary derivative method wit

a game theory approach to mixed h2 h control for a class of stochastic time varying systems with randomly occurring nonlinearities this paper is concerned with the mixed h2 h1 control problem for a class of stochastic time varying systems with nonlinearities the nonlinearities are described by statistical means and could cover several kinds of well studied nonlinearities as special cases the occurrence of the addressed nonlinearities is governed by two sequences of bernoulli distributed white sequences with known probabilities such nonlinearities are named as randomly occurring nonlinearities rons as they appear in a probabilistic way the purpose of the problem under investigation is to design a controller such that the closed loop system achieves the expected h2 performance requirements with a guaranteed h1 disturbance attenuation level a su cient condition is given for the existence of the desired controller by means of the solvability of certain coupled matrix equations by resorting to the game theory approach an algorithm is developed to obtain the controller gain at each sampling instant a numerical example is presented to show the efiectiveness and applicability of the proposed method

finite volume transport schemes we analyze arbitrary order linear finite volume transport schemes and show asymptotic stability in l 1 and l for odd order schemes in dimension one it gives sharp fractional order estimates of convergence for bv solutions it shows odd order finite volume advection schemes are better than even order finite volume schemes therefore the gibbs phenomena is controlled for odd order finite volume schemes numerical experiments sustain the theoretical analysis in particular the oscillations of the lax wendroff scheme for small courant numbers are correlated with its non stability in l 1 a scheme of order three is proved to be stable in l 1 and l

numerical modeling of transient two dimensional viscoelastic waves this paper deals with the numerical modeling of transient mechanical waves in linear viscoelastic solids dissipation mechanisms are described using the generalized zener model no time convolutions are required thanks to the introduction of memory variables that satisfy local in time differential equations by appropriately choosing the relaxation parameters it is possible to accurately describe a large range of materials such as solids with constant quality factors the evolution equations satisfied by the velocity the stress and the memory variables are written in the form of a first order system of pdes with a source term this system is solved by splitting it into two parts the propagative part is discretized explicitly using a fourth order ader scheme on a cartesian grid and the diffusive part is then solved exactly jump conditions along the interfaces are discretized by applying an immersed interface method numerical experiments of wave propagation in viscoelastic and fluid media show the efficiency of this numerical modeling for dealing with challenging problems such as multiple scattering configurations

mixed h sub 2 h sub infinity control a convex optimization approach the problem of finding an internally stabilizing controller that minimizes a mixed h sub 2 h sub infinity performance measure subject to an inequality constraint on the h sub infinity norm of another closed loop transfer function is considered this problem can be interpreted and motivated as a problem of optimal nominal performance subject to a robust stability constraint both the state feedback and output feedback problems are considered it is shown that in the state feedback case one can come arbitrarily close to the optimal even over full information controllers mixed h sub 2 h sub infinity performance measure using constant gain state feedback moreover the state feedback problem can be converted into a convex optimization problem over a bounded subset of n n and n q where n and q are respectively the state and input dimensions real matrices using the central h sub infinity estimator it is shown that the output feedback problem can be reduced to a state feedback problem in this case the dimension of the resulting controller does not exceed the dimension of the generalized plant

ader schemes for the shallow water equations in channel with irregular bottom elevation this paper deals with the construction of high order ader numerical schemes for solving the one dimensional shallow water equations with variable bed elevation the non linear version of the schemes is based on eno reconstructions the governing equations are expressed in terms of total water height instead of total water depth and discharge the eno polynomial interpolation procedure is also applied to represent the variable bottom elevation ader schemes of up to fifth order of accuracy in space and time for the advection and source terms are implemented and systematically assessed with particular attention to their convergence rates non oscillatory results are obtained for discontinuous solutions both for the steady and unsteady cases the resulting schemes can be applied to solve realistic problems characterized by non uniform bottom geometries

numerical modeling of the acoustic wave propagation across a homogenized rigid microstructure in the time domain homogenization of a thin micro structure yields effective jump conditions that incorporate the geometrical features of the scatterers these jump conditions apply across a thin but nonzero thickness interface whose interior is disregarded this paper aims i to propose a numerical method able to handle the jump conditions in order to simulate the homogenized problem in the time domain ii to inspect the validity of the homogenized problem when compared to the real one for this purpose we adapt an immersed interface method originally developed for standard jump conditions across a zero thickness interface doing so allows us to handle arbitrary shaped interfaces on a cartesian grid with the same efficiency and accuracy of the numerical scheme than those obtained in an homogeneous medium numerical experiments are performed to test the properties of the numerical method and to inspect the validity of the homogenization problem

brain venous haemodynamics neurological diseases and mathematical modelling a review behind medicine m is physiology p behind physiology is physics p and behind physics is always mathematics m for which i expect that the symmetry of the quadruplet mppm will be compatible with the characteristic bias of hyperbolic partial differential equations a theme of this paper i start with a description of several idiopathic brain pathologies that appear to have a strong vascular dimension of which the most prominent example considered here is multiple sclerosis the most common neurodegenerative disabling disease in young adults other pathologies surveyed here include retinal abnormalities transient global amnesia transient monocular blindness meniere s disease and idiopathic parkinson s disease it is the hypothesised vascular aspect of these conditions that links medicine to mathematics through fluid mechanics in very complex networks of moving boundary blood vessels the second part of this paper is about mathematical modelling of the human cardiovascular system with particular reference to the venous system and the brain a review of a recently proposed multi scale mathematical model then follows consisting of a one dimensional hyperbolic description of blood flow in major arteries and veins coupled to a lumped parameter description of the remaining main components of the human circulation derivation and analysis of the hyperbolic equations is carried out for blood vessels admitting variable material properties and with emphasis on the venous system a much neglected aspect of cardiovascular mathematics veins unlike their arterial counterparts are highly deformable even collapsible under mild physiological conditions we address mathematical and numerical challenges regarding the numerical analysis of the hyperbolic pdes we deploy a modern non linear finite volume method of arbitrarily high order of accuracy in both space and time the ader methodology in vivo validation examples and brain haemodynamics computations are shown we also point out two preliminary but important new findings through the use of mathematical models namely that extracranial venous strictures produce chronic intracranial venous hypertension and that augmented pressure increases the blood vessel wall permeability

biot jkd model simulation of 1d transient poroelastic waves with fractional derivatives a time domain numerical modeling of biot poroelastic waves is presented the viscous dissipation occurring in the pores is described using the dynamic permeability model developed by johnson koplik dashen jkd some of the coefficients in the biot jkd model are proportional to the square root of the frequency in the time domain these coefficients introduce order 1 2 shifted fractional derivatives involving a convolution product based on a diffusive representation the convolution kernel is replaced by a finite number of memory variables that satisfy local in time ordinary differential equations thanks to the dispersion relation the coefficients in the diffusive representation are obtained by performing an optimization procedure in the frequency range of interest a splitting strategy is then applied numerically the propagative part of biot jkd equations is discretized using a fourth order ader scheme on a cartesian grid whereas the diffusive part is solved exactly comparisons with analytical solutions show the efficiency and the accuracy of this approach

wave simulation in 2d heterogeneous transversely isotropic porous media with fractional attenuation a cartesian grid approach a time domain numerical modeling of transversely isotropic biot poroelastic waves is proposed in two dimensions the viscous dissipation occurring in the pores is described using the dynamic permeability model developed by johnson koplik dashen jkd some of the coefficients in the biot jkd model are proportional to the square root of the frequency in the time domain these coefficients introduce shifted fractional derivatives of order 1 2 involving a convolution product based on a diffusive representation the convolution kernel is replaced by a finite number of memory variables that satisfy local in time ordinary differential equations resulting in the biot da diffusive approximation model the properties of both the biot jkd and the biot da models are analyzed hyperbolicity decrease of energy dispersion to determine the coefficients of the diffusive approximation two approaches are analyzed gaussian quadratures and optimization methods in the frequency range of interest the nonlinear optimization is shown to be the better way of determination a splitting strategy is then applied to approximate numerically the biot da equations the propagative part is discretized using a fourth order ader scheme on a cartesian grid whereas the diffusive part is solved exactly an immersed interface method is implemented to take into account heterogeneous media on a cartesian grid and to discretize the jump conditions at interfaces numerical experiments are presented comparisons with analytical solutions show the efficiency and the accuracy of the approach and some numerical experiments are performed to investigate wave phenomena in complex media such as multiple scattering across a set of random scatterers

modeling 1 d elastic p waves in a fractured rock with hyperbolic jump conditions the propagation of elastic waves in a fractured rock is investigated both theoretically and numerically outside the fractures the propagation of compressional waves is described in the simple framework of 1 d linear elastodynamics the focus here is on the interactions between the waves and fractures for this purpose the mechanical behavior of the fractures is modeled using nonlinear jump conditions deduced from the bandis barton model classically used in geomechanics well posedness of the initial boundary value problem thus obtained is proved numerical modeling is performed by coupling a time domain finite difference scheme with an interface method accounting for the jump conditions the numerical experiments show the effects of contact nonlinearities the harmonics generated may provide a nondestructive means of evaluating the mechanical properties of fractures