new conservative schemes with discrete variational derivatives for nonlinear wave equations new conservative finite difference schemes for certain classes of nonlinear wave equations are proposed the key tool there is discrete variational derivative by which discrete conservation property is realized a similar approach for the target equations was recently proposed by furihata but in this paper a different approach is explored where the target equations are first transformed to the equivalent system representations which are more natural forms to see conservation properties applications for the nonlinear klein gordon equation and the so called good boussinesq equation are presented numerical examples reveal the good performance of the new schemes

analysis of four numerical schemes for a nonlinear klein gordon equation we compare the properties of four explicit finite difference schemes used to integrate the nonlinear klein gordon equation f t t f x x f f 0 it turns out that the energy conserving scheme is the most suitable to study the long time behavior of the solutions the same result is found for the special case of no spatial dependence

the generalized buckley leverett system solvability we propose a new approach to the mathematical modeling of the buckley leverett system which describes two phase flows in porous media considering the initial boundary value problem for a deduced model we prove the solvability of the problem the solvability result relies mostly on the kinetic method

numerical analysis of the cahn hilliard equation with a logarithmic free energy a fully discrete finite element method for the cahn hilliard equation with a logarithmic free energy based on the backward euler method is analysed existence and uniqueness of the numerical solution and its convergence to the solution of the continuous problem are proved two iterative schemes to solve the resulting algebraic problem are proposed and some numerical results in one space dimension are presented

two energy conserving numerical schemes for the sine gordon equation two explicit conservative numerical schemes for the sine gordon equation are proposed their stability and convergence are proved numerical simulation of a sine gordon soliton shows that the new schemes are very accurate and fast

a stable and conservative finite difference scheme for the cahn hilliard equation we propose a stable and conservative finite difference scheme to solve numerically the cahn hilliard equation which describes a phase separation phenomenon numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill posed problem we design a new difference scheme based on a general strategy proposed recently by furihata and mori the new scheme inherits characteristic properties the conservation of mass and the decrease of the total energy from the equation the decrease of the total energy implies boundedness of discretized sobolev norm of the solution this in turn implies by discretized sobolev s lemma boundedness of max norm of the solution and hence the stability of the solution an error estimate for the solution is obtained and the order is o delta x 2 delta t 2 numerical examples demonstrate the effectiveness of the proposed scheme

energy method for structure preserving finite difference schemes and some properties of difference quotient in this article we introduce the energy method for structure preserving finite difference schemes which inherit the physical structures such as energy conservation or dissipative laws another aim is to give some useful properties for difference quotient which is compatible with the structure preserving finite difference schemes the method and properties enable us not only to take the problem with more general nonlinearity but also to improve proofs of error estimate between the numerical and exact solutions lastly we give two examples of application to the cahn hilliard and the boussinesq type equations

preserving energy resp dissipation in numerical pdes using the average vector field method we give a systematic method for discretizing hamiltonian partial differential equations pdes with constant symplectic structure while preserving their energy exactly the same method applied to pdes with constant dissipative structure also preserves the correct monotonic decrease of energy the method is illustrated by many examples in the hamiltonian case these include the sine gordon korteweg de vries nonlinear schrodinger linear time dependent schrodinger and maxwell equations in the dissipative case the examples are the allen cahn cahn hilliard ginzburg landau and heat equations

an energy conservative difference scheme for the nonlinear fractional schrodinger equations abstract in this paper an energy conservative crank nicolson difference scheme for nonlinear riesz space fractional schrodinger equations is studied we give a rigorous analysis of the conservation properties including mass conservation and energy conservation in the discrete sense based on brouwer fixed point theorem the existence of the difference solution is proved by virtue of the energy method the difference solution is shown to be unique and convergent at the order of o 2 h 2 in the l 2 norm with time step and mesh size h finally a linearized iterative algorithm is presented and numerical experiments are given to confirm the theoretical results

symmetric and arbitrarily high order birkhoff hermite time integrators and their long time behaviour for solving nonlinear klein gordon equations abstract the klein gordon equation with nonlinear potential occurs in a wide range of application areas in science and engineering its computation represents a major challenge the main theme of this paper is the construction of symmetric and arbitrarily high order time integrators for the nonlinear klein gordon equation by integrating birkhoff hermite interpolation polynomials to this end under the assumption of periodic boundary conditions we begin with the formulation of the nonlinear klein gordon equation as an abstract second order ordinary differential equation ode and its operator variation of constants formula we then derive a symmetric and arbitrarily high order birkhoff hermite time integration formula for the nonlinear abstract ode accordingly the stability convergence and long time behaviour are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi definite matrix subject to suitable temporal and spatial smoothness a remarkable characteristic of this new approach is that the requirement of temporal smoothness is reduced compared with the traditional numerical methods for pdes in the literature numerical results demonstrate the advantage and efficiency of our time integrators in comparison with the existing numerical approaches

conservative numerical methods for x f x abstract two distinctly different numerical methods are developed for solving the conservative initial value problem x f x x o dot x o both methods conserve exactly the same total energy as does the given differential equation computer examples are described and

numerical simulation of nonlinear schro dinger systems a new conservative scheme we present a new numerical scheme for nonlinear schrodinger type equations the scheme conserves the energy and charge of the systems and it is linearly implicit numerical experiments for several physical problems show that our scheme is stable accurate and efficient

an improved numerical scheme for the sine gordon equation in 2 1 dimensions a rational approximant of order 4 which is applied to a three time level recurrence relation is used to transform the initial boundary value problem associated with the two dimensional sine gordon sg equation arising in the josephson junctions problem the resulting non linear system which is analyzed for stability is solved using an appropriate predictor corrector p c scheme in which an explicit scheme of order 2 is used as predictor for the implementation of the corrector in order to avoid extended matrix evaluations an auxiliary vector was successfully introduced in this p c scheme a modification in the corrector has been proposed according to which the already evaluated corrected values are considered the behavior of this p c scheme is tested numerically on line and ring solitons known from the bibliography regarding the sg equation and conclusions for both undamped and damped problems are derived copyright 2008 john wiley sons ltd

global existence singular solutions and ill posedness for the muskat problem the muskat or muskat leibenzon problem describes the evolution of the interface between two immiscible fluids in a porous medium or hele shaw cell under applied pressure gradients or fluid injection extraction in contras t to the hele shaw problem the one phase version of the muskat problem there are few nontrivial exact solutions or analytic results for the muskat problem for the stable forward muskat problem in which the higher viscosity fluid ex pands into the lower viscosity fluid we show global in time existence for initial data that is a small perturbation of a flat interface the initial data in this result ma y contain weak e g curvature singularities for the unstable backwa rd problem in which the higher viscosity fluid contracts we construct singular solution s that start off with smooth initial data but develop a point of infinite curvature at fi nite time c 2004 wiley periodicals inc

global existence of quasi stratified solutions for the confined ipm equation in this paper we consider a confined physical scenario to prove the global existence of smooth solutions with bounded density and finite energy for the inviscid incompressible porous media ipm equation the result is proved using the stability of stratified solutions combined with an additional structure of our initial perturbation which allows us to get rid of the boundary terms in the energy estimates

finite difference schemes for nonlinear wave equation that inherit energy conservation property abstract we propose two general finite difference schemes that inherit energy conservation property from nonlinear wave equations such as the nonlinear klein gordon equation nlkge one of proposed schemes is implicit and another is explicit many studies exist on fdss that inherit energy conservation property from nlkge and we can derive all of their schemes from the proposed general schemes in this paper the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly because of this the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions we make new fdss for fermi pasta ulam equation string vibration equation shimoji kawai equation ske and ebihara equation and verify numerically the inheritance of the energy conservation property for nlkge and ske

limiting case of the sobolev inequality in bmo with application to the euler equations we shall prove a logarithmic sobolev inequality by means of the bmo norm in the critical exponents as an application we shall establish a blow up criterion of solutions to the euler equations

global well posedness of incompressible flow in porous media with critical diffusion in besov spaces abstract in this paper we study the model of heat transfer in a porous medium with a critical diffusion we obtain global existence and uniqueness of solutions to the equations of heat transfer of incompressible fluid in besov spaces b p 1 3 p r 3 with 1 p by the method of modulus of continuity and fourier localization technique

invariance and existence analysis for semilinear hyperbolic equations with damping and conical singularity abstract in this paper we will discuss about the invariance of solution set and present the existence and non existence of the global solutions a class of initial boundary value problems with dissipative terms is considered for a class of semilinear degenerate hyperbolic equations on the cone sobolev spaces first we will discuss the invariance of some sets corresponding to the problem 1 1 and then by using a family of potential wells and concavity methods we obtain existence and non existence results of global solutions with exponential decay and show the blow up in finite time of solutions on a manifold with conical singularities

a hamiltonian approximation to simulate solitary waves of the korteweg de vries equation given the hamiltonian nature and conservation laws of the korteweg de vries equation the simulation of the solitary waves of this equation by numerical methods should be effected in such a way as to maintain the hamiltonian nature of the problem a semidiscrete finite element approximation of petrov galerkin type proposed by r winther is analyzed here it is shown that this approximation is a finite hamiltonian system and as a consequence the energy integral i u 1 x u3 dx is exactly conserved by this method in addition there is a discussion of error estimates and superconvergence properties of the method in which there is no perturbation term but instead a suitable choice of initial data a single step fully discrete scheme and some numerical results are presented 1 the hamiltonian nature and conservation laws in this paper we shall consider the following problem for the kortewegde vries equation ut 6uux uxxx 0 xer t o p u x 1 t u x t u x 0 uo x a prescribed 1 periodic function to study the hamiltonian nature of problem p we introduce the following function space with i 0 1 h m v e hm i v 0 x v i and the functional h u f x u3 dx received march 24 1989 revised december 7 1989 may 22 1990 1980 mathematics subject classification 1985 revision primary 65n15 65n30

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

a decoupled and conservative difference scheme with fourth order accuracy for the symmetric regularized long wave equations in this paper a decoupled finite difference scheme with fourth order accuracy is proposed to solve the symmetric regularized long wave equation the scheme is proved to conserve the total energy in the discrete level without any restrictions on the grid ratios the convergence of the difference scheme is proved by utilizing the energy method to be of forth order in space and second order in time a numerical example is given to support the theoretical analysis

blow up of solution of cauchy problem for three dimensional damped nonlinear hyperbolic equation abstract the existence and uniqueness of the local generalized solution and the local classical solution to the cauchy problem for three dimensional damped nonlinear hyperbolic equation u t t k 1 4 u k 2 4 u t 2 g 2 u 0 u x 0 u 0 x u t x 0 u 1 x are proved this paper arrives at some sufficient conditions for blow up of the solution in finite time and an example

a note on interface dynamics for convection in porous media we studied the fluid interface problem through porous media given by two incompressible 2 d fluids of different densities this problem is mathematically analogous to the dynamics interface for convection in porous media where the free boundary evolves between fluids with different temperatures we found a new formula for the evolution equation of the free boundary parameterized as a function in the periodic case in this formula there is not a principal value in the non local integral operator involved in the equation giving a simpler system using this formulation we perform numerical simulations in the stable case denser fluid below which shows a strong regularity effect in the periodic interface

growth of solutions for qg and 2d euler equations the work of constantin majda tabak 1 developed an analogy between the quasi geostrophic and 3d euler equations constantin majda and tabak proposed a candidate for a singularity for the quasi geostrophic equation their numerics showed evidence of a blow up for a particular initial data where the level sets of the temperature contain a hyperbolic saddle the arms of the saddle tend to close in finite time producing a sharp front numerics studies done later by ohkitani yamada 8 and constantin nie schorghofer 2 with the same initial data suggested that instead of a singularity the derivatives of the temperature were increasing as double exponential in time the study of collapse on a curve was first studied in 1 for the quasi geostrophic equation where they considered a simplified ansatz for classical frontogenesis with trivial topology at the time of collapse the scalar is discontinuous across the curve x2 f x1 with different limiting values for the temperature on each side of the front they show that under this topology the directional field remains smooth up to the collapse which contradicts the following theorem proven in 1

regular article finite difference schemes for u t x α δg δu that inherit energy conservation or dissipation property we propose a new procedure for designing by rote finite difference schemes that inherit energy conservation or dissipation property from nonlinear partial differential equations such as the korteweg de vries kdv equation and the cahn hilliard equation the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly since the inherited properties are kept even if the time mesh size changes in the time evolution process we can use some appropriate time mesh adaptive methods to obtain numerical solutions through the derived schemes because of these properties the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions and sufficiently flexible to treat the inheritance of the energy conservation and dissipation properties are verified numerically for the kdv equation and the cahn hilliard equation

analytical behavior of two dimensional incompressible flow in porous media in this paper we study the analytic structure of a two dimensional mass balance equation of an incompressible fluid in a porous medium given by darcy s law we obtain local existence and uniqueness by the particle trajectory method and we present different global existence criterions these analytical results with numerical simulations are used to indicate nonformation of singularities moreover blow up results are shown in a class of solutions with infinite energy

numerical integration of the ostrovsky equation based on its geometric structures we consider structure preserving numerical schemes for the ostrovsky equation which describes gravity waves under the influence of coriolis force this equation has two associated invariants an energy function and the l 2 norm it is widely accepted that structure preserving methods such as invariants preserving and multi symplectic integrators generally yield qualitatively better numerical results in this paper we propose five geometric integrators for this equation energy preserving and norm preserving finite difference and galerkin schemes and a multi symplectic integrator based on a newly found multi symplectic formulation a numerical comparison of these schemes is provided which indicates that the energy preserving finite difference schemes are more advantageous than the other schemes

analyticity of lagrangian trajectories for well posed inviscid incompressible fluid models abstract we discuss general incompressible inviscid models including the euler equations the surface quasi geostrophic equation incompressible porous medium equation and boussinesq equations all these models have classical unique solutions at least for short time we show that they have real analytic lagrangian paths more precisely we show that as long as a solution of any of these equations is in a class of regularity that assures holder continuous gradients of velocity the corresponding lagrangian paths are real analytic functions of time the method of proof is conceptually straightforward and general and we address the combinatorial issues head on

general local energy preserving integrators for solving multi symplectic hamiltonian pdes in this paper we propose and investigate a general approach to constructing local energy preserving algorithms which can be of arbitrarily high order in time for solving hamiltonian pdes this approach is based on the temporal discretization using continuous runge kutta type methods and the spatial discretization using pseudospectral methods or gauss legendre collocation methods the local energy conservation law of our new schemes is analyzed in detail the effectiveness of the novel local energy preserving integrators is demonstrated by coupled nonlinear schrodinger equations and 2d nonlinear schrodinger equations with external fields our new schemes are compared with some classical multi symplectic and symplectic schemes in numerical experiments the numerical results show the remarkable long term behaviour of our new schemes

on the initial and terminal value problem for a class of semilinear strongly material damped plate equations abstract we study the semilinear strongly damped plate equation by considering its two different problems for initial value problem we prove the local well posedness and blow up results of solution for the problem with polynomial nonlinear source terms for terminal value problem given the ill posedness in the sense of hadamard we propose a regularization method for the problem with logarithmic nonlinearities under the a priori assumption on the exact solution belonging to a gevrey space we propose the fourier truncation method to stabilize the ill posed problem also by establishing some stability estimates of logarithmic type in l q space

remarks on energy methods for structure preserving finite difference schemes small data global existence and unconditional error estimate abstract in the previous article yoshikawa 2017 the author proposes the energy method for structure preserving finite difference schemes which enable us to show global existence and uniqueness of solution for the schemes and error estimates in this article we give two extended remarks of the methods one is related to the small data global existence results for schemes of which energy is not necessarily bounded from below the other is an unconditional error estimate which holds globally in time and without smallness condition for split sizes these results can be shown due to the structure preserving property

a general framework for deriving integral preserving numerical methods for pdes a general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented in particular linearly implicit methods preserving a time discretized version of the invariant are developed for systems of partial differential equations with polynomial nonlinearities the framework is rather general and allows for an arbitrary number of dependent and independent variables with derivatives of any order it is proved formally that second order convergence is obtained the procedure is applied to a test case and numerical experiments are provided

on a stress resultant geometrically exact shell model part vi conserving algorithms for non linear dynamics in parts i to v of the present work the formulation and finite element implementation of a non linear stress resultant shell model are considered in detail this paper is concerned with the extension of these results to incorporate completely general non linear dynamic response of special interest here is the dynamics of very flexible shells undergoing large overall motion which conserves the total linear and angular momentum and for the hamiltonian case the total energy a main goal of this paper is the design of non linear time stepping algorithms and the construction of finite element interpolations which preserve exactly these fundamental constants of motion it is shown that only a very special class of algorithms namely a formulation of the mid point rule in conservation form exactly preserves the total linear and angular momentum for the hamiltonian case a somewhat surprising result is proved regardless of the degree of non linearity in the stored energy function a generalized mid point rule algorithm always exists which exactly conserves energy the conservation properties of a time stepping algorithm need not and in general will not be preserved by the spatial discretization precise conditions which ensure preservation of these conservation properties are derived a number of numerical simulations are presented which illustrate the exact conservation properties of the proposed methodology

finite difference calculus invariant structure of a class of algorithms for the nonlinear klein gordon equation in a previous work the authors have presented a formalism for deriving systematically invariant symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities this formalism is herein cast in the context of exact finite difference calculus the algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus in the same sense as that of the corresponding differential equation being derivable from its associated energy function a conserved quantity a clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities which are the consistent counterpart of the invariant quantities in the continuous case results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism and that preserve energy or linear momentum are discussed in the context of finite difference calculus some

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

scalars convected by a two dimensional incompressible flow we provide a test for numerical simulations for several two dimensional incompressible flows that appear to develop sharp fronts we show that in order to have a front the velocity has to have uncontrolled velocity growth 2001 john wiley sons inc

lack of uniqueness for weak solutions of the incompressible porous media equation in this work we consider weak solutions of the incompressible two dimensional porous media ipm equation by using the approach of de lellis szekelyhidi we prove non uniqueness for solutions in l in space and time

geometric statistics in turbulence the author presents results regarding certain average properties of incompressible fluids derived from the equations of motion the author estimates the average dissipation rate the average dimension of level sets the role played by the field of direction of vorticity in the three dimensional euler and navier stokes equations is discussed and a class of two dimensional equations that are useful models of incompressible dynamics is described the author presents results concerning scaling exponents in turbulence

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells