short note a new minimum storage runge kutta scheme for computational acoustics a new fourth order six stage runge kutta numerical integrator that requires 2n storage n is the number of degrees of freedom of the system with low dissipation and dispersion and a relatively large stability interval is proposed these features make it a suitable time advancing method for solving wave propagation problems in computational acoustics some numerical experiments are presented to show the favourable behaviour of the new scheme as compared with the ldd46 and ldd25 methods proposed by stanescu and habashi j comput phys 143 1998 674 and the standard fourth order runge kutta method

on the decay of solutions for porous elastic systems with history in this paper we study the asymptotic behavior to an one dimensional porous elasticity problem with history we show the lack of exponential stability when the porous dissipation or the elastic dissipation is absent and we show the lack of analyticity and exponential stability when the porous viscosity and the elastic dissipation are present

energy decay for timoshenko systems of memory type abstract linear systems of timoshenko type equations for beams including a memory term are studied the exponential decay is proved for exponential kernels while polynomial kernels are shown to lead to a polynomial decay the optimality of the results is also investigated

a note on iterated splitting schemes iterated splittings seem attractive in view of consistency and local accuracy in this note it will be shown however that for stiff systems the stability properties are quite poor specific runge kutta implementations can improve stability but this leads to classes of methods that are better studied in their own right

uncertainty propagation using infinite mixture of gaussian processes and variational bayesian inference uncertainty propagation in flow through porous media problems is a challenging problem this is due to the high dimensionality of the random property fields e g permeability and porosity as well as the computational complexity of the models that are involved the usual approach is to construct a surrogate response surface and then use it instead of the expensive model to carry out the uncertainty propagation task however the construction of the surrogate surface is hampered by various aspects such as the limited number of model evaluations that one can afford the curse of dimensionality multi variate responses with non trivial correlations potential localized features of the response and or discontinuities in this work we extend upon the concept of the multi output gaussian process mgp to effectively deal with all of these difficulties simultaneously this non trivial extension involves an infinite mixture of mgp s that is trained using variational bayesian inference prior to observing any data a dirichlet process is used to generate the components of the mgp mixture the bayesian nature of the model allows for the quantification of the uncertainties due to the limited number of simulations i e we can derive error bars for the statistics of interest the automatic detection of the mixture components by the variational inference algorithm is able to capture discontinuities and localized features without adhering to ad hoc constructions finally correlations between the components of multi variate responses are captured by the underlying mgp model in a natural way

richardson extrapolated sequential splitting and its application during numerical time integration the accuracy of the numerical solution obtained with a given step size often proves unsatisfactory in this case one usually reduces the step size and repeats the computation while the results obtained for the coarser grid are not used however we can also combine the two solutions and obtain a better result this idea is based on the richardson extrapolation a general technique for increasing the order of an approximation method this technique also allows us to estimate the absolute error of the underlying method in this paper we apply richardson extrapolation to the sequential splitting and investigate the performance of the resulting scheme on several test examples

implicit solution of the unsteady euler equations for high order accurate discontinuous galerkin discretizations efficient solution techniques for high order accurate time dependent problems are investigated for solving the two dimensional non linear euler equations in this work the spatial discretization consists of a high order accurate discontinuous galerkin dg approach implicit time integration techniques are considered exclusively in order to avoid the stability restrictions of explicit methods standard backwards differencing methods bdf1 and bdf2 as well as a second order crank nicholson cn2 and a fourth order implicit runge kutta irk4 scheme are considered in an attempt to balance the spatial and temporal accuracy of the overall approach the implicit system arising at each time step is solved using a p multigrid approach which is shown to produce h independent convergence rates while remaining relatively insensitive to the time step size the crank nicholson methodology although not l stable demonstrates superior performance compared to the bdf2 scheme for the problems chosen in this work however the fourth order accurate implicit runge kutta scheme is found to be the most efficient in terms of computational cost for a given accuracy level as compared to the lower order schemes in spite of the added cost per time step and the benefits of this scheme increase for tighter error tolerances

analyticity to transmission problem with delay in porous elasticity abstract in this paper we consider a one dimensional transmission problem in a bounded domain with a delay in porous elasticity using a semigroup theorem under suitable assumption on the weight of the delay term we establish the well posedness of the system furthermore using the method developed by z liu and s zheng we show that the semigroup associated with the dissipative system is analytic and consequently exponentially stable

poisson deformations of symplectic quotient singularities abstract we establish a connection between smooth symplectic resolutions and symplectic deformations of a possibly singular affine poisson variety in particular let v be a finite dimensional complex symplectic vector space and g sp v a finite subgroup our main result says that the so called calogero moser deformation of the orbifold v g is in an appropriate sense a versal poisson deformation that enables us to determine the algebra structure on the cohomology h x c of any smooth symplectic resolution x v g multiplicative mckay correspondence we prove further that if g gl h is an irreducible weyl group and v h h then no smooth symplectic resolution of v g exists unless g is of types a b c

runge kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics in this paper a new runge kutta method with minimal dispersion and dissipation error is developed the chebyshev pseudospectral method is utilized using spatial discretization and a new fourth order six stage runge kutta scheme is used for time advancing the proposed scheme is more efficient than the existing ones for acoustic computations

runge kutta methods for hyperbolic conservation laws with stiff relaxation terms underresolved numerical schemes for hyperbolic conservation laws with stiff relaxation terms may generate unphysical spurious numerical results or reduce to lower order if the small relaxation time is not temporally well resolved we design a second order runge kutta type splitting method that possesses the discrete analogue of the continuous asymptotic limit which thus is able to capture the correct physical behaviors with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved

stabilization in elastic solids with voids in this paper we study the asymptotic behavior and the analyticity of the solutions of the one dimensional porous elasticity problem with thermal effect our main result is to prove the lack of exponential stability in case of the porous elasticity with thermal effect when viscoelasticity is present we prove the analyticity of the problem when a porous viscosity is present we conclude by showing the impossibility of localization in time of the solutions in the isothermal case

symplectic resolutions for nilpotent orbits in this paper firstly we calculate picard groups of a nilpotent orbit in a classical complex simple lie algebra and discuss the properties of being factorial and factorial for the normalization tilde of the closure of then we consider the problem of symplectic resolutions for tilde our main theorem says that for any nilpotent orbit in a semi simple complex lie algebra equipped with the kostant kirillov symplectic form if for a resolution z tilde the 2 form defined on 1 extends to a symplectic 2 form on z then z is isomorphic to the cotangent bundle t g p of a projective homogeneous space and is the collapsing of the zero section it proves a conjecture of cho miyaoka shepherd barron in this special case using this theorem we determine all varieties tilde which admit such a resolution

on richardson extrapolation for low dissipation low dispersion diagonally implicit runge kutta schemes abstract in the modeling of wave propagation phenomena it is necessary to use time integration methods which are not only sufficiently accurate but also properly describe the amplitude and phase of the propagating waves it is not clear if amending the developed schemes by extrapolation methods to obtain a high order of accuracy preserves the qualitative properties of these schemes in the perspective of dissipation dispersion and stability analysis it is illustrated that the combination of various optimized schemes with richardson extrapolation is not optimal for minimal dissipation and dispersion errors optimized third order and fourth order methods are obtained and it is shown that the proposed methods combined with richardson extrapolation result in fourth and fifth orders of accuracy correspondingly while preserving optimality and stability the numerical applications include the linear wave equation a stiff system of reaction diffusion equations and the nonlinear euler equations with oscillatory initial conditions it is demonstrated that the extrapolated third order scheme outperforms the recently developed fourth order diagonally implicit runge kutta scheme in terms of accuracy and stability

the convergence of diagonally implicit runge kutta methods combined with richardson extrapolation runge kutta methods are widely used in the solution of systems of ordinary differential equations richardson extrapolation is an efficient tool for enhancing the accuracy of time integration schemes in this paper we investigate the convergence of the combination of any of the diagonally implicit including also the explicit runge kutta methods with active richardson extrapolation and show that the numerical solution obtained converges under rather natural conditions

operator splitting methods in respect of eigenvalue problems for nonlinear equations and applications for burgers equations in this article we consider iterative operator splitting methods for nonlinear differential equations with respect to their eigenvalues the main focus of the proposed idea is the fixed point iterative scheme that linearizes our underlying equations on the basis of the approximated eigenvalues of such linearized systems we choose the order of the operators for our iterative splitting scheme the convergence properties of such a mixed method are studied and demonstrated we confirm with numerical applications the effectiveness of the proposed scheme in comparison with the standard operator splitting methods by providing improved results and convergence rates we apply our results to deposition processes

additive and iterative operator splitting methods and their numerical investigation as an alternative to the classical splitting methods two new splitting schemes have been developed recently the additive and the iterative splitting in this paper we discuss the most important properties the advantages and disadvantages of these schemes and investigate their performance on simple examples as well as on more complex physical models

operator splitting for chemical reaction systems with fast chemistry abstract splitting methods are frequently used for large scale chemical reaction systems the main reason is the reduced computational cost for solving the subproblems in comparison to the time integration of the full problem however a splitting error is introduced the most popular splitting schemes are the first order lie trotter splitting and the second order strang splitting however in case of stiff differential equations the strang splitting suffers from order reduction and both schemes have order one for stiff differential equations therefore the step size restrictions due to the low order can result in a prohibitive small step size hence splitting schemes with order larger than one are necessary for stiff differential equations the richardson extrapolation of the lie trotter splitting is a second order scheme in this paper we examine differential equations with a fast chemical source term and a slow transport term thus all stiffness of the system is related to the chemical source term we show that the richardson extrapolation does not suffer from order reduction in this case furthermore we perform a stability analysis of the extrapolated splitting scheme for a linear test problem thereby the operators of the test problem do not commute

general decay of solutions of a linear one dimensional porous thermoelasticity system with a boundary control of memory type abstract in this paper we consider linear porous thermoelasticity systems in a bounded domain where the memory type damping is acting on a part of the boundary we establish a general decay result for which the usual exponential and polynomial decay rates are just special cases our work allows certain relaxation functions which are not necessarily of exponential or polynomial decay and therefore generalizes and improves on earlier results from the literature

semi implicit runge kutta schemes development and application to compressible channel flow time stepconstraintdueto thefine meshnear thewall the explicitsubschemeis asix stage fourth order low storage runge kutta scheme based on analysis in fourier space and results obtained for propagation test cases the semiimplicitschemesareshowntobeoforder3and forwavesdiscretizedbyanumberofpointsperperiodbetween4and 16 tobeasaccurateas ormoreaccuratethan thestandardexplicitfourth orderrunge kuttaalgorithmintermsof dissipationanddispersion thelarge eddysimulationofacompressibleturbulentchannelflowatafrictionreynolds numberof360andamachnumberof0 1isthencarriedoutwithoneoftheproposedalgorithms thecomputational time is reduced by a factor 1 33 with respect to a large eddy simulation using the explicit subscheme in all directions wall pressure and velocity spectra from the large eddy simulation are presented to give insights into the flow turbulent structures in particular wave number frequency spectra are calculated acoustic components appear to be identified in these spectra

induced nilpotent orbits and birational geometry abstract in general a nilpotent orbit closure in a complex simple lie algebra g does not have a crepant resolution but it always has a q factorial terminalization by the minimal model program according to b fu a nilpotent orbit closure has a crepant resolution only when it is a richardson orbit and the resolution is obtained as a springer map for it in this paper we shall generalize this result to q factorial terminalizations when g is classical here the induced orbits play an important role instead of richardson orbits

birational geometry of symplectic quotient singularities for a finite subgroup gamma subset mathrm sl 2 mathbb c and for n ge 1 we use variation of git quotient for nakajima quiver varieties to study the birational geometry of the hilbert scheme of n points on the minimal resolution s of the kleinian singularity mathbb c 2 gamma it is well known that x mathrm mathrm hilb n s is a projective crepant resolution of the symplectic singularity mathbb c 2n gamma n where gamma n gamma wr mathfrak s n is the wreath product we prove that every projective crepant resolution of mathbb c 2n gamma n can be realised as the fine moduli space of theta stable pi modules for a fixed dimension vector where pi is the framed preprojective algebra of gamma and theta is a choice of generic stability condition our approach uses the linearisation map from git to relate wall crossing in the space of theta stability conditions to birational transformations of x over mathbb c 2n gamma n as a corollary we describe completely the ample and movable cones of x over mathbb c 2n gamma n and show that the mori chamber decomposition of the movable cone is determined by an extended catalan hyperplane arrangement of the ade root system associated to gamma by the mckay correspondence in the appendix we show that morphisms of quiver varieties induced by variation of git quotient are semismall generalising a result of nakajima in the case where the quiver variety is smooth

on the structure of homogeneous symplectic varieties of complete intersection a normal complex algebraic variety x is called a symplectic variety if its regular locus xreg admits a holomorphic symplectic 2 form such that it extends to a holomorphic 2 form on a resolution f x x affine symplectic varieties are constructed in various ways such as nilpotent orbit closures of a semisimple complex lie algebra cf cm slodowy slices to nilpotent orbits cf sl or symplectic reductions of holomorphic symplectic manifolds with hamiltonian actions usually these examples come up with c actions in this article we shall study a 2n dimensional affine symplectic variety x c defined as a complete intersection of r homogeneous polynomials fi z1 z2n r 1 i r here we assume that weights of all coordinates are 1 wt z1 wt z2n r 1 the c action on c induces a c action on x we also assume that the symplectic form is homogeneous with respect to thisc action namely for some integer l we have t t where t c the integer l is called the weight of and is denoted by wt a main result theorem 2 is that such an x is isomorphic as a cvariety to a nilpotent orbit closure of a semisimple complex lie algebra and corresponds to the kostant kirillov form on o a nilpotent orbit closure does not generally have complete intersection singularities and the orbit that appears in theorem 2 should be very restricted one we conjecture that such a nilpotent orbit closure would be the nilpotent variety of a semisimple complex lie algebra as a particular case we consider a symplectic hypersurface i e r 1 as is studied in lnsv we have very few examples of quasihomogeneous symplectic hypersurfaces in our homogeneous case it exists only when n 1 and x must be isomorphic to an a1 surface singularity theorem 3

deformation theory of singular symplectic n folds by a symplectic manifold or a symplectic n fold we mean a compact kaehler manifold of even dimension n with a non degenerate holomorphic 2form i e is a nowhere vanishing n form this notion is generalized to a variety with singularities we call x a projective symplectic variety if x is a normal projective variety with rational gorenstein singularities and if the regular locus u of x admits a non degenerate holomorphic 2 form a symplectic variety will play an important role together with a singular calabi yau variety in the generalized bogomolov decomposition conjecture now that essentially a few examples of symplectic manifolds are discovered it seems an important task to seek new symplectic manifolds by deforming symplectic varieties in this paper we shall study a projective symplectic variety from a view point of deformation theory if x has a resolution x x such that x is a symplectic manifold we say that x has a symplectic resolution our first results are concerned with a birational contraction map of a symplectic manifold

a new linear quotient of c 4 admitting a symplectic resolution we show that the quotient c4 g admits a symplectic resolution for g q 8 times bf z 2 d 8 sf sp 4 bf c here q8 is the quaternionic group of order eight and d8 is the dihedral group of order eight and g is the quotient of their direct product which identifies the nontrivial central elements id of each it is equipped with the tensor product representation bf c 2 boxtimes bf c 2 cong bf c 4 this group is also naturally a subgroup of the wreath product group q 8 2 rtimes s 2 sf sp 4 bf c we compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming c4 g we also discuss preliminary investigations on the more general question of classifying linear quotients v g admitting symplectic resolutions

high order low dissipation low dispersion diagonally implicit runge kutta schemes high order low dissipation low dispersion diagonally implicit runge kutta schemes are analyzed and introduced based on the optimization of amplification and phase errors for wave propagation various optimized schemes can be obtained the new scheme shows negligible dissipation it is illustrated mathematically and numerically that the new scheme preserves fourth order accuracy while the recently developed diagonally implicit runge kutta scheme does not the numerical applications contain the advection equation with and without a stiff nonlinear source term and an oscillatory test the new scheme is a stable as desired for the solution of stiff problems

general decay of solutions in one dimensional porous elastic system with memory abstract in this work we consider a one dimensional porous elastic system with memory effects it is well known that porous elastic system with a single dissipation mechanism lacks exponential decay in contrary we prove that the unique dissipation given by the memory term is strong enough to exponentially stabilize the system depending on the kernel of the memory term and the wave speeds of the system in fact we prove a general decay result for which exponential and polynomial decay results are special cases our result is new and improves previous results in the literature

on symplectic resolutions and factoriality of hamiltonian reductions recently herbig schwarz seaton have shown that 3 large representations of a reductive group g give rise to a large class of symplectic singularities via hamiltonian reduction we show that these singularities are always terminal we show that they are mathbb q factorial if and only if g has finite abelianization when g is connected and semi simple we show they are actually locally factorial as a consequence the symplectic singularities do not admit symplectic resolutions when g is semi simple we end with some open questions

high order embedded runge kutta formulae abstract the criteria to be satisfied by embedded runge kutta pairs of formulae are reviewed two new formulae of orders 6 and 8 are presented together with tests on their efficiency relative to other high order formulae in current use

an analysis of operator splitting techniques in the stiff case operator splitting methods are commonly used in many applications we focus here on the case where the evolution equations to be simulated are stiff we will more particularly consider the case of two operators a stiff one and a nonstiff one this occurs in numerous application fields e g combustion air pollution and reactive flows the classical analysis of the splitting error may then fail since the chosen splitting timestep t is in practice much larger than the fastest time scales the asymptotic expansion t 0 is therefore no longer valid we show here that singular perturbation theory provides an interesting framework for the study of splitting error some new results concerning the order of local errors are derived the main result deals with the choice of the sequential order for the operators the stiff operator must always be last in the splitting scheme

weighted sequential splittings and their analysis operator splitting is a widely used procedure in the numerical solution of parabolicproblems several splitting methods have been constructed and used in the fields of applied mathematics in this paper the weighted sequential splittings are introduced and their local splitting error is analyzed new results are presented for the case where the subproblems of the weighted splitting schemes are solved numerically the results are illustrated with test examples for ordinary differential systems

on the time polynomial decay in elastic solids with voids in this paper we investigate the temporal decay behavior of the solutions of the one dimensional problem in various theories of continua with voids it has been proved that the coupling of the elastic structure with porous microstructure is weak in the sense that in many situations the temporal decay of solutions is slow we have considered some theories of porous continua when the deformation rate tensor or time rate or porosity function or thermal effects is present we have proved that the decay cannot be controlled by a negative exponential the natural question now is whether there exist or not a polynomial rate of decay of the solution in some appropriate norms in this paper we consider some cases where the decay is slow and we obtain polynomial decay estimates in concrete we consider the case when only the viscoelastic effect is present the case when the motion of voids is assumed to be quasi static and the porous viscosity is present and we finish with the case of the porous elasticity when thermal effect is coupled

diagonally implicit runge kutta nystro m methods for oscillatory problems implicit runge kutta nystrom rkn methods are constructed for the integration of second order differential equations possessing an oscillatory solution based on a linear homogeneous test model we analyse the phase errors or dispersion introduced by these methods and derive so called dispersion relations diagonally implicit rkn methods of relatively low algebraic order are constructed which have a high order of dispersion up to 10 application of these methods to a number of test examples linear as well as nonlinear yields a greatly reduced phase error when compared with conventional dirkn methods

general decay rates for one dimensional porous elastic system with memory the case of non equal wave speeds abstract feng and apalara 2019 16 investigated the one dimensional porous elastic system with finite memory under the assumptions of equal speed wave propagations and positive definite energy associated with the solution and established an optimal explicit energy decay rate in this paper we consider the case of non equal wave speeds and positive semidefinite energy which are more realistic from the physics point of view by using the second order energy we prove some new decay results that generalize and improve many earlier results in the literature

optimized explicit finite difference schemes for spatial derivatives using maximum norm we propose an optimized scheme using the maximum norm and the simulated annealing the maximum norm offers the largest set of possible solutions for solvers to search the explicit finite difference operator is greatly improved by our optimized scheme we use a tight error limitation to make accuracy improvement to be high and solid our optimized scheme allows greater saving of computational cost and memory demand conventional explicit finite difference methods have difficulties in handling high frequency components due to strong numerical dispersions one can reduce the numerical dispersions by optimizing the constant coefficients of the finite difference operator different from traditional optimized schemes that use the 2 norm and the least squares we propose to construct the objective functions using the maximum norm and solve the objective functions using the simulated annealing algorithm both theoretical analyses and numerical experiments show that our optimized scheme is superior to traditional optimized schemes with regard to the following three aspects first it provides us with much more flexibility when designing the objective functions thus we can use various possible forms and contents to make the objective functions more reasonable second it allows for tighter error limitation which is shown to be necessary to avoid rapid error accumulations for simulations on large scale models with long travel times finally it is powerful to obtain the optimized coefficients that are much closer to the theoretical limits which means greater savings in computational efforts and memory demand

optimal decay for a porous elasticity system with memory abstract in this paper we consider a one dimensional linear porous system with finite memory effective on the equilibrated stress vector by assuming minimal conditions on the relaxation function we establish an optimal explicit and energy decay rate for which exponential and polynomial rates are special cases this result is new and substantially improves earlier results in the literature

a family of low dispersive and low dissipative explicit schemes for flow and noise computations explicit numerical methods for spatial derivation filtering and time integration are proposed they are developed with the aim of computing flow and noise with high accuracy and fidelity all the methods are constructed in the same way by minimizing the dispersion and the dissipation errors in the wavenumber space up to k x 2 corresponding to four points per wavelength they are shown to be more accurate and also more efficient numerically than most of the standard explicit high order methods for uniform and slowly non uniform grids two problems involving long range sound propagation are resolved to illustrate their respective precisions remarks about their practical applications are then made especially about the connection with the boundary conditions finally their relevance for the simulation of turbulent flows is emphasized

a noumerov type method with minimal phase lag for the integration of second order periodic initial value abstract in a recent paper 2 we gave a noumerov type method with minimal phase lag for the integration of second order initial value problems y f t y y t0 y0 y t0 y 0 however the method given there is implicit we show here the interesting result that if the noumerov type methods of 2 are made explicit with the help of the classical second order method then there exists a selection of the free parameter for which the resulting method has a considerably small frequency distortion of size 1 40320 h 6 and also a slightly larger interval of periodicity of size 2 75 than the phase lag of size 1 12096 h 6 and interval of periodicity of size 2 71 for the implicit method 2 more interestingly it turns out that noumerov made explicit of chawla 3 also has less frequency distortion than the implicit noumerov method we shall assume a familiarity with the notation and discussion given in 2

a semi implicit mid point rule for stiff systems of ordinary differential equations the paper introduces a new semi implicit extrapolation method especially designed for the numerical solution of stiff systems of ordinary differential equations the existence of a quadratic asymptotic expansion in terms of the stepsize is shown moreover the new discretization is analyzed in the light of well known stability models the efficiency of the new integrator is clearly demonstrated by solving a series of challenging test problems including real life examples