high resolution schemes for conservation laws with locally varying time steps we develop upwind methods which use limited high resolution corrections in the spatial discretization and local time stepping for forward euler and second order time discretizations l infty stability is proven for both time stepping schemes for problems in one space dimension these methods are restricted by a local cfl condition rather than the traditional global cfl condition allowing local time refinement to be coupled with local spatial refinement numerical evidence demonstrates the stability and accuracy of the methods for problems in both one and two space dimensions

hybrid fourier pseudospectral discontinuous galerkin time domain method for wave propagation the fourier pseudospectral time domain fourier pstd method was shown to be an efficient way of modelling acoustic propagation problems as described by the linearized euler equations lee but is limited to real valued frequency independent boundary conditions and predominantly staircase like boundary shapes this paper presents a hybrid approach to solve the lee coupling fourier pstd with a nodal discontinuous galerkin dg method dg exhibits almost no restrictions with respect to geometrical complexity or boundary conditions the aim of this novel method is to allow the computation of complex geometries and to be a step towards the implementation of frequency dependent boundary conditions by using the benefits of dg at the boundaries while keeping the efficient fourier pstd in the bulk of the domain the hybridization approach is based on conformal meshes to avoid spatial interpolation of the dg solutions when transferring values from dg to fourier pstd while the data transfer from fourier pstd to dg is done utilizing spectral interpolation of the fourier pstd solutions the accuracy of the hybrid approach is presented for one and two dimensional acoustic problems and the main sources of error are investigated it is concluded that the hybrid methodology does not introduce significant errors compared to the fourier pstd stand alone solver an example of a cylinder scattering problem is presented and accurate results have been obtained when using the proposed approach finally no instabilities were found during long time calculation using the current hybrid methodology on a two dimensional domain a method for wave propagation in fluid domains with complex geometries is presented the method combines fourier pstd in the bulk of the domain and dg at boundaries the main application of interest is atmospheric sound propagation in cities the method shows no significant additional error compare with a fourier pstd solver no indication of instability in long time calculations has been detected

a local refinement finite element method for one dimensional parabolic systems vector systems of parabolic partial differential equations in one space dimension are solved by an adaptive local mesh refinement galerkin finite element procedure piecewise linear polynomials are used for the spatial representation of the solution and the backward euler method is used for temporal integration a local error indicator based on an estimate of the local discretization error is used to control an adaptive feedback mesh refinement strategy where finer space time meshes are recursively added to coarser ones in regions where greater solution resolution is needed a posteriori estimates of the local discretization error are obtained by a p refinement procedure that uses piecewise quadratic hierarchic finite element approximations in space and trapezoidal rule integration in time superconvergence properties of the finite element method are used to neglect errors at nodes and thereby increase the computational efficiency of the error estimation procedure further improvements in computational e

binary nonlinearization of the super akns system under an implicit symmetry constraint for the super akns system an implicit symmetry constraint between the potentials and the eigenfunctions is proposed after introducing some new variables to explicitly express potentials the super akns system is decomposed into two compatible finite dimensional super systems x part and tn part furthermore we show that the obtained super systems are integrable super hamiltonian systems in the supersymmetry manifold

dispersive and dissipative behaviour of high order discontinuous galerkin finite element methods the dispersive and dissipative properties of hp version discontinuous galerkin finite element approximation are studied in three different limits for the small wave number limit hk 0 we show the discontinuous galerkin gives a higher order of accuracy than the standard galerkin procedure thereby confirming the conjectures of hu and atkins j comput phys 182 2 2002 516 if the mesh is fixed and the order p is increased it is shown that the dissipation and dispersion errors decay at a super exponential rate when the order p is much larger than hk finally if the order is chosen so that 2p 1 hk for some fixed constant 1 then it is shown that an exponential rate of decay is obtained

explicit finite element methods for symmetric hyperbolic equations a family of explicit space time finite element methods for the initial boundary value problem for linear symmetric hyperbolic systems of equations is described and analyzed the method generalizes the discontinuous galerkin method and as is typical for this method obtains error estimates of order o h n 1 2 for approximations by polynomials of degree le n

a posteriori error estimation for discontinuous galerkin solutions of hyperbolic problems abstract we analyze the spatial discretization errors associated with solutions of one dimensional hyperbolic conservation laws by discontinuous galerkin methods dgms in space we show that the leading term of the spatial discretization error with piecewise polynomial approximations of degree p is proportional to a radau polynomial of degree p 1 on each element we also prove that the local and global discretization errors are o x2 p 1 and o x2p 1 at the downwind point of each element this strong superconvergence enables us to show that local and global discretization errors converge as o xp 2 at the remaining roots of radau polynomial of degree p 1 on each element convergence of local and global discretization errors to the radau polynomial of degree p 1 also holds for smooth solutions as p these results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors that are effective for linear and nonlinear conservation laws in regions where solutions are smooth

asymptotic trichotomy for positive solutions of a class of odd order nonlinear neutral difference equations abstract a class of neutral type nonlinear difference equations is studied we show that eventually positive solutions either converge to zero to positive constants or to positive infinity sufficient conditions are then provided for the existence of these solutions partial converses are also given

quadrature free implementation of discontinuous galerkin method for hyperbolic equations a discontinuous galerkin formulation that avoids the use of discrete quadrature formulas is described and applied to linear and nonlinear test problems in one and two space dimensions this approach requires less computational time and storage than conventional implementations but preserves the compactness and robustness inherent in the discontinuous galerkin method test problems include the linear and nonlinear one dimensional scalar advection of smooth initial value problems that are discretized by using unstructured grids with varying degrees of smoothness and regularity and two dimensional linear euler solutions on unstructured grids

a trace identity and its applications to the theory of discrete integrable systems a general theory for studying discrete integrable systems is developed based on a trace identity that the author previously proposed a scheme for generating hierarchies of discrete integrable systems is presented under broad assumptions the resulting hierarchies are shown to consist of liouville integrable hamiltonian systems

an integrable coupling family of the toda lattice systems its bi hamiltonian structure and a related nonisospectral integrable lattice family starting from a four by four matrix spectral problem an integrable coupling family of the toda lattice systems is derived a pair of discrete hamiltonian operators is presented and a sequence of the corresponding hamiltonian functionals is constructed by using the discrete variational identity then the bi hamiltonian structure of the obtained integrable coupling family is established finally a nonisospectral integrable lattice family associated with the obtained integrable coupling family is given through nonisospectral discrete zero curvature representation

on unbounded positive solutions of second order difference equations with a singular nonlinear term abstract in this paper several necessary and or sufficient conditions are given for the existence of unbounded positive solutions of nonlinear difference equations of the form r n x n f n x n 0 n n n 0 where n n0 n0 n0 1 n0 is a fixed nonnegative integer rn 0 for n n0 such that n n0 1 rn and f n n0 r r is a real valued continuous function such that f n x 0 for all n x n n0 0

r matrix approach to lattice integrable systems an r matrix formalism is applied to the construction of the integrable lattice systems and their bi hamiltonian structure miura like gauge transformations between the hierarchies are also investigated in the end the ladder of linear maps between generated hierarchies is established and described

adaptive local overlapping grid methods for parabolic systems in two space dimensions abstract adaptive mesh refinement techniques are described for two dimensional systems of parabolic partial differential equations solutions are calculated using galerkin s method with a piecewise bilinear basis in space and backward euler integration in time a posteriori estimates of the local discretization error of piecewise bilinear finite element solutions are obtained by a p refinement technique these error estimates are used to control a local h refinement strategy where finer grids are recursively introduced in regions where a prescribed tolerance is exceeded fine grids at a given level of refinement may overlap each other and independent solutions are generated on each of them a version of the schwarz alternating principle is used to coordinate solutions between overlapping fine grids computational results demonstrating the performance of the adaptive procedure on linear and nonlinear problems and apparent convergence of the error estimate for linear heat conduction problem and uniform global refinement is presented

a high order immersed boundary method for acoustic wave scattering and low mach number flow induced sound in complex geometries a new sharp interface immersed boundary method based approach for the computation of low mach number flow induced sound around complex geometries is described the underlying approach is based on a hydrodynamic acoustic splitting technique where the incompressible flow is first computed using a second order accurate immersed boundary solver this is followed by the computation of sound using the linearized perturbed compressible equations lpce the primary contribution of the current work is the development of a versatile high order accurate immersed boundary method for solving the lpce in complex domains this new method applies the boundary condition on the immersed boundary to a high order by combining the ghost cell approach with a weighted least squares error method based on a high order approximating polynomial the method is validated for canonical acoustic wave scattering and flow induced noise problems applications of this technique to relatively complex cases of practical interest are also presented

a family of liouville integrable lattice equations with a parameter and its two symmetry constraints a two by two matrix spectral problem with an arbitrary constant is introduced a family of integrable lattice equations involving a parameter is derived by the technique of lax pairs the typical lattice equation in the obtained family is a deformed toda lattice system the bi hamiltonian structure of the obtained family of integrable lattice equations is established by using the discrete trace identity and we prove the liouville integrability of obtained family of integrable lattice equations then bargmann symmetry constraint and an implicit symmetry constraint of the obtained family are presented by binary nonlinearization method under these two symmetry constraints we obtain two series of integrable symplectic maps and two sequences of completely integrable finite dimensional hamiltonian systems in liouville sense and every integrable lattice equation in the obtained family is decomposed by an integrable symplectic map and a completely integrable finite dimensional hamiltonian system as a specia

curved boundary treatments for the discontinuous galerkin method applied to aeroacoustic propagation a discontinuous galerkin method is applied to unstructured grids to simulate aeroacoustic propagation modeled by the linearized euler equations on triangular and tetrahedral elements with straight edges the quadrature free form of the discontinuous galerkin method is used in addition to the classical linear treatment of wall boundaries two treatments involving a second order representation of the geometry are presented the simulation of acoustic scattering problems shows that the linear treatment can limit the accuracy at high order and demonstrates how the boundary treatment involving curved elements overcomes this restriction the benefits of higher order treatments are also assessed for more realistic geometries namely a high lift airfoil and an elliptical muffler

symmetries conserved quantities and hierarchies for some lattice systems with soliton structure basic invariants such as conserved quantities symmetries mastersymmetries and recursion operators are explicitly constructed for the following nonlinear lattice systems the modified korteweg de vries lattice the ablowitz ladik lattice the brusci ragnisco lattice the ragnisco tu lattice and some cases of the class of integrable systems introduced by bogoyavlensky the algorithmic basis for obtaining these quantities is described and the interrelation between the underlying mastersymmetry approach and the lax pair analysis is discussed by explicit presentation of the higher order members of the corresponding hierarchies new completely integrable lattice flows are found for all systems multi hamiltonian formulations are given

a modified toda spectral problem and its hierarchy of bi hamiltonian lattice equations starting from a modified toda spectral problem a hierarchy of generalized toda lattice equations with two arbitrary constants is constructed through discrete zero curvature equations it is shown that the hierarchy possesses a bi hamiltonian structure and a hereditary recursion operator which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals two cases of the involved constants present two specific integrable sub hierarchies one of which is exactly the toda lattice hierarchy

binary symmetry constraints of n wave interaction equations in 1 1 and 2 1 dimensions binary symmetry constraints of the n wave interaction equations in 1 1 and 2 1 dimensions are proposed to reduce the n wave interaction equations into finite dimensional liouville integrable systems a new involutive and functionally independent system of polynomial functions is generated from an arbitrary order square matrix lax operator and used to show the liouville integrability of the constrained flows of the n wave interaction equations the constraints on the potentials resulting from the symmetry constraints give rise to involutive solutions to the n wave interaction equations and thus the integrability by quadratures are shown for the n wave interaction equations by the constrained flows

integrable couplings of relativistic toda lattice systems in polynomial form and rational form their hierarchies and bi hamiltonian structures integrable couplings of relativistic toda lattice systems in polynomial form and rational form and their hierarchies are derived from a four by four discrete matrix eigenvalue problem the bi hamiltonian structure for every integrable coupling in the two hierarchies obtained is established by means of the discrete variational identity ultimately liouvolle integrability of the obtained integrable couplings is demonstrated

optimal numerical parameterization of discontinuous galerkin method applied to wave propagation problems this paper deals with the high order discontinuous galerkin dg method for solving wave propagation problems first we develop a one dimensional dg scheme and numerically compute dissipation and dispersion errors for various polynomial orders an optimal combination of time stepping scheme together with the high order dg spatial scheme is presented it is shown that using a time stepping scheme with the same formal accuracy as the dg scheme is too expensive for the range of wave numbers that is relevant for practical applications an efficient implementation of a high order dg method in three dimensions is presented using 1d convergence results we further show how to adequately choose elementary polynomial orders in order to equi distribute a priori the discretization error we also show a straightforward manner to allow variable polynomial orders in a dg scheme we finally propose some numerical examples in the field of aero acoustics