Numerical methods of Lagrangian points and particles for one-dimensional wave gas-dynamics equations

In this paper, an approach is formulated to construct numerical methods of computational gas dynamics based on the approximation of the second-order nonlinear wave equations (NWE) in time and spatial variables. The NWE approach enables the construction of finite-difference and finite-element schemes with the balance (conservation) cells both in the form of finite volumes and Lagrangian points and particles. That is why the numerical methods based on the NWE approximation are of great interest for the solution of both one-dimensional (1D) and multidimensional problems of gas dynamics. In this work, we construct and study discrete NWE models for 1D problems of gas dynamics in Lagrangian variables and discuss the results of numerical experiments.     

Chebyshev rational spectral method for long-short wave equations

We consider the initial boundary value problem of the long-short wave equations on the whole line. A fully discrete spectral approximation scheme is developed based on Chebyshev rational functions in space and central difference in time. A priori estimates are derived which are crucial to study numerical stability and convergence of the fully discrete scheme. Then, unconditional numerical stability is proved. Convergence of the fully discrete scheme is shown by the method of error estimates. Finally, numerical experiments are presented to demonstrate the efficiency and accuracy of the convergence results.     

A flux splitting method for the numerical simulation of one-dimensional compressible flow

Using the technique of flux vector splitting, it is shown that one-dimensional, inviscid, compressible-flow equations possess a split conservation form. Some attractive features of this form for the design of finite-difference solution schemes are discussed. Based on the split form, two solution chemes are designed. One is a first-order accurate ‘upwind’ scheme and the other is similar to the Lax-Wendroff scheme. A hybrid scheme, based on a nonlinear weighting of these two schemes, is demonstrated to yield results superior to either of the two in the solution of an ideal shock-tube problem.     

On a finite element method for the equations of one-dimensional nonlinear thermoviscoelasticity

A computationally uncoupled numerical scheme for the equations of one-dimensional nonlinear thermoviscoelasticity is proposed. The scheme makes use of the finite element method for the space variable and the different method for the time variable. The existence and uniqueness of the approximate solutions are proved, and bounds of the error are analyzed.     

New characteristic difference method with adaptive mesh for one-dimensional unsteady convection-dominated diffusion equations

This article presents a kind of characteristic difference method with adaptive mesh for one-dimensional convection-dominated diffusion equations. The method can adaptively adjust the computational mesh according to the gradient of the solutions. Furthermore, the method in this article has second order accuracy for convective term and first order accuracy for diffusive term, which completely fits with the property of strong convection and weak diffusion for convection-dominated diffusion equations. Compared with the uniform mesh method, the method in this article has higher computational efficiency for linear and nonlinear convection-dominated diffusion equations.     

An application of the decomposition method for second order wave equations

In this paper we study the solution of a linear and nonlinear damped wave and dissipative wave equations by Adomian decomposition method. We illustrate that the analytic solutions and a reliable numerical approximation of the damped wave and dissipative wave equations are calculated in the form of a series with easily computable components. The nonhomogeneous problem is quickly solved by observing the self-canceling"noise"terms whose sum vanishes in the limit. In comparison to traditional techniques, the series based technique of Adomian decomposition method is shown to evaluate solutions accurately and cheaply.     

Accuracy improvement of a multistep splitting method for nonlinear viscous wave equations

This paper focuses on a multistep splitting method for a class of nonlinear viscous equations in two spaces, which uses second-order backward differentiation formula (BDF2) combined with approximation factorization for time integration, and second-order centred difference approximation to second derivatives for spatial discretization. By the discrete energy method, it is shown that this splitting method can attain second-order accuracy in both time and space with respect to the discrete L²- and H¹-norms. Moreover, for improving computational efficiency, we introduce a Richardson extrapolation method and obtain extrapolation solution of order four in both time and space. Numerical experiments illustrate the accuracy and performance of our algorithms.     

Classification of conservation laws for KdV-like equations

We prove that the computation of the conservation laws of (2n + 1)th-order KdV-like equations (i.e. higher order evolution equations with the same scaling as KdV) can be restricted to polynomials with constant terms, except when the order of the conservation law equals n − 1, in which case the density has linear t dependence. This shows that existing computer algebra programs which assume the conservation law to be of this form are providing the complete answer.     

An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations

The tanh-function method for finding explicit travelling solitary wave solutions to non-linear evolution equations is described. The method is usually extremely tedious to use by hand. We present a Mathematica package ATFM that deals with the tedious algebra and outputs directly the required solutions. The use of the package is illustrated by applying it to a variety of equations; not only are previously known solutions recovered but in some cases more general forms of solution are obtained.     

A symplectic algorithm for wave equations

Numerical schemes for finite-dimensional Hamiltonian system which preserve the symplectic structure are generalized to infinite-dimensional Hamiltonian systems and applied to construct finite difference schemes for the nonlinear wave equation. The numerical results show that these schemes compare favorably with conventional difference methods. Furthermore, the successful long-term tracking capability for these Hamiltonian schemes is remarkable and striking.     

Kinetic flux vector splitting for radiation hydrodynamical equations

This paper is interested in the kinetic flux vector splitting (KFVS) for the multidimensional radiation hydrodynamical equations (RHEs) in zero diffusion limit. First, a generalized Maxwell–Boltzmann distribution function with two new parameters of temperature approximation is introduced to recover the macroscopic equations. These parameters are uniquely determined by macroscopic variables. Then, a high resolution KFVS method is proposed for the solution of the multidimensional RHEs. It does not require any Riemann solvers. Finally, several numerical examples are given to show the performance of our scheme.     

A new all-speed flux scheme for the Euler equations

Since being proposed, the HLLEM-type schemes have been widely used because they are with high discontinuity resolutions and can be easily applied to the other system of hyperbolic conservation law. In this paper, we conduct theoretical analyses on the HLLE-type schemes’ performances at low speeds. By realizing that the excessive numerical dissipations corresponding to the velocity-difference terms of the momentum equations make these schemes incapable of obtaining physical solutions at low speeds, we adopt the function $g$ to control such dissipation. Also, we borrow the HLLEMS scheme’s construction and damp the shear waves in the vicinity of the shock to avoid the shock anomaly’s appearance. The moving contact discontinuity case and the Sod shock tube case show that the HLLEMS-AS scheme we propose in this paper can capture contact discontinuities and shocks as sharply as HLLEMS scheme. The Quirk’s odd–even test case and the hypersonic inviscid flow over a cylinder case demonstrate that HLLEMS-AS is robust against the shock anomaly. The inviscid low-speed flow around the NACA0012 airfoil case indicates that HLLEMS-AS is with a high resolution at low speeds. The turbulent flow past a backward facing step case demonstrates the shear wave capturing ability of the HLLEMS-AS scheme. These properties suggest that HLLEMS-AS is promising to be widely used in both cases of low speed and high speed.     

Optimal design for 2D wave equations

In this paper We consider a problem of optimal design in 2D for the wave equation with Dirichlet boundary conditions. We introduce a finite element discrete version of this problem in which the domains under consideration are polygons defined on the numerical mesh. We prove that, as the mesh size tends to zero, any limit, in the sense of the complementary-Hausdorff convergence, of discrete optimal shapes is an optimal domain for the continuous optimal design problem. We work in the functional and geometric setting introduced by V. ?veràk in which the domains under consideration are assumed to have an a priori limited number of holes. We present in detail a numerical algorithm and show the efficiency of the method through various numerical experiments.     

Arbitrarily wide-angle wave equations for complex media

By combining various ideas related to one-way wave equations (OWWEs), half-space stiffness relation, special finite-element discretization, and complex coordinate stretching, a systematic procedure is developed for deriving a series of highly accurate space-domain versions of OWWEs. The resulting procedure is applicable to complex media where the governing equation (full wave equation) is a second order differential system, making the procedure applicable for general heterogeneous, anisotropic, porous, viscoelastic media. Owing to their high accuracy in representing waves propagating in an arbitrarily wide range of angles, the resulting equations are named Arbitrarily Wide-angle Wave Equations (AWWEs). In order to illustrate the proposed procedure, AWWEs are derived for one-way propagation in acoustic as well as elastic media. While acoustic AWWEs can be considered as modified versions of well-known space-domain OWWEs based on rational approximations of the square root operator, the elastic AWWEs are significantly different from the existing elastic OWWEs. Unlike the existing elastic OWWEs, elastic AWWEs are displacement-based and are applicable to general anisotropic media. Furthermore, AWWEs are simple in their form, and appear amenable to easy numerical implementation.     

Multiscale-stabilized solutions to one-dimensional systems of conservation laws

We present a variational multiscale formulation for the numerical solution of one-dimensional systems of conservation laws. The key idea of the proposed formulation, originally presented by Hughes [Comput. Methods Appl. Mech. Engrg., 127 (1995) 387–401], is a multiple-scale decomposition into resolved grid scales and unresolved subgrid scales. Incorporating the effect of the subgrid scales onto the coarse scale problem results in a finite element method with enhanced stability properties, capable of accurately representing the sharp features of the solution. In the formulation developed herein, the multiscale split is invoked prior to any linearization of the equations. Special attention is given to the choice of the matrix of stabilizing coefficients and the discontinuity-capturing diffusion. The methodology is applied to the one-dimensional simulation of three-phase flow in porous media, and the shallow water equations. These numerical simulations clearly show the potential and applicability of the formulation for solving highly nonlinear, nearly hyperbolic systems on very coarse grids. Application of the numerical formulation to multidimensional problems is presented in a forthcoming paper.     

Multisymplecticity and wave action conservation

This paper discuses some novel results concerning the wave action conservation law for multisymplectic partial differential equations and their discretizations. We provide a method for deriving this conservation law in Fourier spectral space. A discrete wave action conservation law for a multisymplectic box scheme and for the midpoint time-discretization of a spectral method is also derived.     

Numerical study of the one-dimensional coupled nonlinear sine-Gordon equations by a novel geometric meshless method

Engineering with Computers - In the paper, we derive a geometric meshless method for coupled nonlinear sine-Gordon (CNSG) equations. Approximate solutions of the CNSG equations are supposed to be...     

New solitary wave solutions to nonlinear evolution equations by the Exp-function method

A new application of the Exp-function method in combination with the dependent variable transformation from singularity analysis is proposed for constructing new generalized solitary wave solutions and periodic wave solutions for nonlinear evolution equations. The Korteweg–de Vries equation is chosen to illustrate the validity and applicability of the suggested approach.     

Numerical solutions of non-linear kinetic equations for a one-dimensional evaporation-condensation problem

Numerical solutions have been obtained for both the nonlinear Boltzmann equation (for two collision laws) and the Krook equation for a one-dimensional evaporation-condensation problem for a range of parameters. Our calculations of rate indicate that the linear prediction underestimates the non-equilibrium hindrance effect due to intermolecular collisions. The Krook evaporation rates are lower than the corresponding Boltzmann results and, thus, differ even more from the linear values. The Boltzmann evaporation rates for a gas of hard spheres are lower than those for Maxwelian molecules at lower values of Knudsen number. The microscopic and macroscopic properties obtained from the Krook solutions differ appreciably from the corresponding Boltzmann results.     

Iterative method for fuzzy equations

In this paper, we propose numerical solution for solving a system of fuzzy nonlinear equations based on Fixed point method. The convergence theorem is proved in detail. In this method the algorithm is illustrated by solving several numerical examples.