High order conservative Lagrangian schemes with Lax–Wendroff type time discretization for the compressible Euler equations

In this paper, we explore the Lax–Wendroff (LW) type time discretization as an alternative procedure to the high order Runge–Kutta time discretization adopted for the high order essentially non-oscillatory (ENO) Lagrangian schemes developed in 3 and 5. The LW time discretization is based on a Taylor expansion in time, coupled with a local Cauchy–Kowalewski procedure to utilize the partial differential equation (PDE) repeatedly to convert all time derivatives to spatial derivatives, and then to discretize these spatial derivatives based on high order ENO reconstruction. Extensive numerical examples are presented, for both the second-order spatial discretization using quadrilateral meshes [3] and third-order spatial discretization using curvilinear meshes [5]. Comparing with the Runge–Kutta time discretization procedure, an advantage of the LW time discretization is the apparent saving in computational cost and memory requirement, at least for the two-dimensional Euler equations that we have used in the numerical tests.     

A central Rankine–Hugoniot solver for hyperbolic conservation laws

A numerical method in which the Rankine–Hugoniot condition is enforced at the discrete level is developed. The simple format of central discretization in a finite volume method is used together with the jump condition to develop a simple and yet accurate numerical method free of Riemann solvers and complicated flux splittings. The steady discontinuities are captured accurately by this numerical method. The basic idea is to fix the coefficient of numerical dissipation based on the Rankine–Hugoniot (jump) condition. Several numerical examples for scalar and vector hyperbolic conservation laws representing the inviscid Burgers equation, the Euler equations of gas dynamics, shallow water equations and ideal MHD equations in one and two dimensions are presented which demonstrate the efficiency and accuracy of this numerical method in capturing the flow features.     

A nonlinear instability for 3×3 systems of conservation laws

The phenomenon of nonlinear resonance provides a mechanism for the unbounded amplification of small solutions of systems of conservation laws. We construct spatially 2-periodic solutionsu ^N C ([0,t _N ] × witht _N bounded, satisfying

Soliton solutions and conservation laws for a generalized Ablowitz–Ladik system

In this paper, a generalized Ablowitz–Ladik system is systemically investigated via the Darboux transformation method. Soliton solutions and conservation laws are presented. Depending on the choices of parameters, the dynamic behaviors are discussed graphically.     

Symmetries and conservation laws of one Blaszak–Marciniak four-field lattice equation

In this paper, by using the classical Lie symmetry approach, Lie point symmetries and reductions of one Blaszak– Marciniak(BM) four-field lattice equation are obtained. Two kinds of exact solutions of a rational form and an exponential form are given. Moreover, we show that the equation has a sequence of generalized symmetries and conservation laws of polynomial form, which further confirms the integrability of the BM system.     

Infinitely many conservation laws for the Blaszak–Marciniak four-field integrable lattice hierarchy

In this Letter, by means of new matrix Lax representation for the Blaszak–Marciniak four-field integrable lattice hierarchy, we demonstrate the existence of infinitely many conservation laws for the lattice hierarchy and give the corresponding conserved density and the associated flux formulaically. So, its integrability is further confirmed.     

Invariant solutions and conservation laws of the generalized Kaup–Boussinesq equation

The generalized Kaup–Boussinesq equation is a model which is used to describe the water wave. In this paper, Lie group analysis method is used to perform detailed analysis on the generalized Kaup–Boussinesq equation. Some invariant solutions are obtained under the transformation groups. The conservation laws of the generalized Kaup–Boussinesq equation are constructed using two methods: multiplier method and Ibragimov theorem.     

An efficient method for the incompressible Navier–Stokes equations on irregular domains with no-slip boundary conditions,high order up to the boundary

Common efficient schemes for the incompressible Navier–Stokes equations, such as projection or fractional step methods, have limited temporal accuracy as a result of matrix splitting errors, or introduce errors near the domain boundaries (which destroy uniform convergence to the solution). In this paper we recast the incompressible (constant density) Navier–Stokes equations (with the velocity prescribed at the boundary) as an equivalent system, for the primary variables velocity and pressure. equation for the pressure. The key difference from the usual approaches occurs at the boundaries, where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time. This avoids the common difficulty of an, apparently, over-determined Poisson problem. Since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity, the recast equations are ideal for numerical marching methods. The new system can be discretized using a variety of methods, including semi-implicit treatments of viscosity, and in principle to any desired order of accuracy. In this work we illustrate the approach with a 2-D second order finite difference scheme on a Cartesian grid, and devise an algorithm to solve the equations on domains with curved (non-conforming) boundaries, including a case with a non-trivial topology (a circular obstruction inside the domain). This algorithm achieves second order accuracy in the L^∞ norm, for both the velocity and the pressure. The scheme has a natural extension to 3-D.     

Numerical method satisfying the first two conservation laws for the Korteweg–de Vries equation

In this paper, we develop a finite-volume scheme for the KdV equation which conserves both the momentum and energy. The main ingredient of the method is a numerical device we developed in recent years that enables us to construct numerical method for a PDE that also simulates its related equations. In the method, numerical approximations to both the momentum and energy are conservatively computed. The operator splitting approach is adopted in constructing the method in which the conservation and dispersion parts of the equation are alternatively solved; our numerical device is applied in solving the conservation part of the equation. The feasibility and stability of the method is discussed, which involves an important property of the method, the so-called Jensen condition. The truncation error of the method is analyzed, which shows that the method is second-order accurate. Finally, several numerical examples, including the Zabusky–Kruskal’s example, are presented to show the good stability property of the method for long-time numerical integration.     

Solitons and conservation laws in magneto–optic waveguides having parabolic–nonlocal law of refractive index

This paper reveals soliton solutions to magneto–optic waveguides that maintain parabolic–nonlocal law of refractive index. The unified Riccati equation expansion together with extended auxiliary equation approach together reveal bright, dark, singular as well as straddled optical solitons. These soliton solutions are obtained through a limiting process when the modulus of ellipticity approaches unity. Finally, the conservation laws are also listed.     

Combination of high order polarization vortex beams using Hermite–Gauss beams

A method to combine arbitrary order polarization vortex beams (PVBs) by Hermite–Gauss mode beams with designed field distributions is proposed. The orders of PVBs are determined by the modes of HG beams which are used in the combination scheme and the intensity distributions are the same as those of Laguerre–Gaussian beams. Simulations and experiments were also demonstrated, which verifies the feasibility of the proposed method.     

Covariantized Nœther identities and conservation laws for perturbations in metric theories of gravity

A construction of conservation laws and conserved quantities for perturbations in arbitrary metric theories of gravity is developed. In an arbitrary field theory, with the use of incorporating an auxiliary metric into the initial Lagrangian covariantized N?ther identities are carried out. Identically conserved currents with corresponding superpotentials are united into a family. Such a generalized formalism of the covariantized identities gives a natural basis for constructing conserved quantities for perturbations. A new family of conserved currents and correspondent superpotentials for perturbations on arbitrary curved backgrounds in metric theories is suggested. The conserved quantities are both of pure canonical N?ther and of Belinfante corrected types. To test the results each of the superpotentials of the family is applied to calculate the mass of the Schwarzschild-anti-de Sitter black hole in the Einstein–Gauss–Bonnet gravity. Using all the superpotentials of the family gives the standard accepted mass.     

Solutions and conservation laws of Benjamin–Bona–Mahony–Peregrine equation with power-law and dual power-law nonlinearities

In this paper, exact solutions of Benjamin–Bona–Mahony–Peregrine equation are obtained with power-law and dual power-law nonlinearities. The Lie group analysis as well as the simplest equation method are used to carry out the integration of these equations. The solutions obtained are cnoidal waves, periodic solutions and soliton solutions. Subsequently, the conservation laws are derived for the underlying equations.     

Nonclassical Lie symmetry and conservation laws of the nonlinear time-fractional Korteweg–de Vries equation

In this paper, we use the symmetry of the Lie group analysis as one of the powerful tools that deals with the wide class of fractional order differential equations in the Riemann–Liouville concept. In this study, first, we employ the classical and nonclassical Lie symmetries(LS) to acquire similarity reductions of the nonlinear fractional far field Korteweg–de Vries(KdV)equation, and second, we find the related exact solutions for the derived generators. Finally,according to the LS generators acquired, we construct conservation laws for related classical and nonclassical vector fields of the fractional far field Kd V equation.     

Spectral parameter dependent Lax pairs for systems of Calogero–Moser type

Presentation of a method for generating Lax pairs for systems obtained by means of Hamiltonian reduction.     

Lie symmetry analysis,conservation laws and analytic solutions of the time fractional Kolmogorov–Petrovskii–Piskunov equation

In this paper, we consider the invariance properties of the multiple-term fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation. By employing the Lie symmetry analysis method, we explicitly investigate the vector fields and symmetry reductions of the FKPP equation. Moreover, an effective method is proposed to succinctly derive the exact power series solutions with their convergence analysis of the equation. Finally, by using the new conservation theorem, the conservation laws associated with Lie symmetries of the equation are well constructed with a detailed analysis.     

A stable and conservative high order multi-block method for the compressible Navier–Stokes equations

A stable and conservative high order multi-block method for the time-dependent compressible Navier–Stokes equations has been developed. Stability and conservation are proved using summation-by-parts operators, weak interface conditions and the energy method. This development makes it possible to exploit the efficiency of the high order finite difference method for non-trivial geometries. The computational results corroborate the theoretical analysis.     

An Efficient Method for the Solution of Schwinger–Dyson Equations for Propagators

Efficient computation methods are devised for the perturbative solution of Schwinger–Dyson equations for propagators. I show how a simple computation allows to obtain the dominant contribution in the sum of many parts of previous computations. This allows for an easy study of the asymptotic behavior of the perturbative series. In the cases of the four-dimensional supersymmetric Wess–Zumino model and the f₆³{\phi_6^3} complex scalar field, the singularities of the Borel transform for both positive and negative values of the parameter are obtained and compared.     

Comparison between lattice Boltzmann method and Navier–Stokes high order schemes for computational aeroacoustics

Computational aeroacoustic (CAA) simulation requires accurate schemes to capture the dynamics of acoustic fluctuations, which are weak compared with aerodynamic ones. In this paper, two kinds of schemes are studied and compared: the classical approach based on high order schemes for Navier–Stokes-like equations and the lattice Boltzmann method. The reference macroscopic equations are the 3D isothermal and compressible Navier–Stokes equations. A Von Neumann analysis of these linearized equations is carried out to obtain exact plane wave solutions. Three physical modes are recovered and the corresponding theoretical dispersion relations are obtained. Then the same analysis is made on the space and time discretization of the Navier–Stokes equations with the classical high order schemes to quantify the influence of both space and time discretization on the exact solutions. Different orders of discretization are considered, with and without a uniform mean flow. Three different lattice Boltzmann models are then presented and studied with the Von Neumann analysis. The theoretical dispersion relations of these models are obtained and the error terms of the model are identified and studied. It is shown that the dispersion error in the lattice Boltzmann models is only due to the space and time discretization and that the continuous discrete velocity Boltzmann equation yield the same exact dispersion as the Navier–Stokes equations. Finally, dispersion and dissipation errors of the different kind of schemes are quantitatively compared. It is found that the lattice Boltzmann method is less dissipative than high order schemes and less dispersive than a second order scheme in space with a 3-step Runge–Kutta scheme in time. The number of floating point operations at a given error level associated with these two kinds of schemes are then compared.