High-order Conservative Scheme Research Articles

High Order Conservative Schemes for the Generalized Benjamin–Ono Equation on the Unbounded Domain

High Order Conservative Schemes for the Generalized Benjamin–Ono Equation on the Unbounded Domain

High-order conservative schemes for the nonlinear Schrödinger equation in the semiclassical limit

This letter devotes to the design of efficient prediction–correction numerical methods which produce high-order approximations of the solutions while preserving mass, or energy, or both of them, for the semiclassical Schrödinger equation with small Planck constant ɛ. The prediction step involves an explicit temporal fourth-order exponential Runge–Kutta method which allows the ɛ-oscillatory solution to be captured efficiently. The correction step only requires solving algebraic nonlinear equations. Numerical results show that the present methods have good meshing strategies τ=O(ɛ) and h=O(ɛ) and excellent power in the simulation of Bose–Einstein condensation.

Error Estimates of High-order Conservative Schemes for the Generalized Nonlinear Schrödinger Equation in One Dimension

In this paper, we use the high-order difference operator Dμ to compensate the central difference operator . Then, we discretize time by the time-midpoint method. Fully discrete schemes are obtained for the generalized nonlinear Schrödinger equation. For the high-order methods, conservation of mass and energy can be proved theoretically in the discrete sense. Then, to establish the optimal error estimates without any requirement on the grid ratio, we adopt the cut-off function technique. The error estimates of the schemes are proved to be of O(τ2 + h2(μ+1)) with time step size τ and mesh size h. Some numerical experiments are given to verify the accuracy order. In addition, the dynamics of the generalized nonlinear Schrödinger equation in one dimension are simulated.

High-order linearly implicit schemes conserving quadratic invariants

In this paper, we propose linearly implicit and arbitrary high-order conservative numerical schemes for ordinary differential equations with a quadratic invariant. Many differential equations have invariants, and numerical schemes for preserving them have been extensively studied. Since linear invariants can be easily kept after discretisation, quadratic invariants are essentially the simplest ones. Quadratic invariants are important objects that appear not only in many physical examples but also in the computationally efficient conservative schemes for general invariants such as scalar auxiliary variable approach, which have been studied in recent years. It is known that quadratic invariants can be maintained relatively easily compared with general invariants, and can be preserved by canonical Runge–Kutta methods. However, there is no unified method for constructing linearly implicit and high order conservative schemes. In this paper, we construct such schemes based on canonical Runge–Kutta methods and prove some properties involving accuracy.

Arbitrary high-order exponential integrators conservative schemes for the nonlinear Gross-Pitaevskii equation

In this paper, we propose a family of high-order conservative schemes based on the exponential integrators technique and the symplectic Runge-Kutta method for solving the nonlinear Gross-Pitaevskii equation. By introducing generalized scalar auxiliary variable, the equation is equivalent to a new system with both mass and modified energy conservation laws. Then, a conservative semi-discrete exponential scheme is given by combining the symplectic Runge-Kutta method and the Lawson method in the time direction. Subsequently, we apply the Fourier pseudo-spectral method to approximate semi-discrete system in space and obtain the fully-discrete schemes that conserve the energy and mass. Numerical examples are presented to confirm the accuracy and conservation of the developed schemes.

High-order conservative scheme for the coupled space fractional nonlinear Schrödinger equations

In this paper, an efficient finite difference scheme is proposed for one dimension and two dimension coupled space fractional nonlinear Schrödinger equations. First, the high-order difference scheme and Crank–Nicolson scheme are used to one dimension coupled space fractional nonlinear Schrödinger equations. second, we show that the high-order conservative difference scheme satisfies the mass and energy conservation laws respectively, and convergence and unconditional stability of the scheme are also proved. Next, we give the high-order conservative scheme for two dimension coupled space fractional nonlinear Schrödinger equations. Finally, some numerical results are reported to verify our theoretical analysis.

High-order conservative schemes for the space fractional nonlinear Schrödinger equation

In the paper, the high-order conservative schemes are presented for space fractional nonlinear Schrödinger equation. First, we give two class high-order difference schemes for fractional Risze derivative by compact difference method and extrapolating method, and show the convergence analysis of the two methods. Then, we apply high-order conservative difference schemes in space direction, and Crank-Nicolson, linearly implicit and relaxation schemes in time direction to solve fractional nonlinear Schrödinger equation. Moreover, we show that the arising schemes are uniquely solvable and approximate solutions converge to the exact solution at the rate O(τ2+h4), and preserve the mass and energy conservation laws. Finally, we given numerical experiments to show the efficiency of the conservative finite difference schemes.

Implicit fast sweeping method for hyperbolic systems of conservation laws

Implicit time-accurate methods are often used to integrate stiff problems where explicit schemes impose severe time step restrictions. This paper presents an efficient numerical framework based on the fast sweeping method (FSM) for solving linear and nonlinear hyperbolic systems of conservation laws. The solution at each discrete location is computed by sweeping the numerical domain in several predetermined directions that follow the causality of the characteristic families. The use of a fractional step strategy eliminates the need for a solution selection criterion while one-sided stencils limit the number of sweeps to at most 2d for d space dimensions. This work focuses on the first-order implicit upwind method since it constitutes the building block for high-order conservative schemes. For problems where the degree of stiffness evolves over time, implicit-explicit hybridization can be accomplished with the same algorithm by simply switching the stencil at each time level. As opposed to traditional implicit solvers, the sweeping method does not require a local time linearization of the fluxes thereby preserving the nonlinear stability properties of the original implicit scheme. It also avoids the large computational and memory requirements associated with solving large block-diagonal systems of equations. A series of one- and two-dimensional test cases are presented for the inviscid Burgers' equation and the reactive Euler equations. The results indicate that the implicit FSM can allow for a major reduction in the number of time steps even in the presence of discontinuous solution profiles.

High-order conservative schemes for the nonlinear Dirac equation

ABSTRACT The fine physical details for the quantum computation can easily be provided by high-order conservative schemes. By compensating the high-order difference operator to the central difference operator δ, a time-dependent semi-discrete system, which conserves both charge and energy, is derived for the two-dimensional nonlinear Dirac equation (NLDE). Two kinds of fully discrete schemes are obtained by discretizing this semi-discrete system in time with the time-midpoint method and the time-splitting method. We prove theoretically that the former one conserves both charge and energy while the latter one only keeps the charge conservation. Some numerical experiments are given to verify the accuracy order, the stability and the conservative properties. In addition, the dynamics of the NLDE in one dimension and two dimensions are simulated.

Semi-Conservative Finite Volume Schemes for Conservation Laws

This paper aims to introduce a new class of high order conservative schemes to solve systems of conservation laws. The idea is to couple the conservation form of the system with, possibly simpler, ...

Positivity-preserving method for high-order conservative schemes solving compressible Euler equations

In this work a simple method to enforce the positivity-preserving property for general high-order conservative schemes is proposed for solving compressible Euler equations. The method detects critical numerical fluxes which may lead to negative density and pressure, and for such critical fluxes imposes a simple flux limiter by combining the high-order numerical flux with the first-order Lax–Friedrichs flux to satisfy a sufficient condition for preserving positivity. Though an extra time-step size condition is required to maintain the formal order of accuracy, it is less restrictive than those in previous works. A number of numerical examples suggest that this method, when applied on an essentially non-oscillatory scheme, can be used to prevent positivity failure when the flow involves vacuum or near vacuum and very strong discontinuities.