Conservative Numerical Scheme Research Articles

A high-accuracy conservative numerical scheme for the generalized nonlinear Schrödinger equation with wave operator

<p>In this article, we establish a novel high-order energy-preserving numerical approximation scheme to study the initial and periodic boundary problem of the generalized nonlinear Schrödinger equation with wave operator, which is proposed by the finite difference method. The scheme is of fourth-order accuracy in space and second-order one in time. The conservation property of energy as well as a priori estimate are described. The convergence of the proposed scheme is discussed in detail by using the energy method. Some comparisons have been made between the proposed method and the others. Numerical examples are presented to illustrate the validity and accuracy of the method.</p>

A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation

Abstract In this article, a numerical solution for the Rosenau-Kawahara equation is considered. A new conservative numerical approximation scheme is presented to solve the initial boundary value problem of the Rosenau-Kawahara equation, which preserves the original conservative properties. The proposed scheme is based on the finite difference method. The existence of the numerical solutions for the scheme has been shown by Browder fixed point theorem. The priori bound and error estimates, as well as the conservation of discrete mass and discrete energy for the finite difference solutions, are discussed. The discrepancies of discrete mass and energy are computed and shown by the curves of these quantities over time. Unconditional stability, second-order convergence, and uniqueness of the scheme are proved based on the discrete energy method. Numerical examples are given to show the effectiveness of the proposed scheme and confirm the theoretical analysis.

Implicit Gradients Based Conservative Numerical Scheme for Compressible Flows

This paper introduces a novel approach to compute the numerical fluxes at the cell boundaries for a cell-centered conservative numerical scheme. Explicit gradients used in deriving the reconstruction polynomials are replaced by high-order gradients computed by compact finite differences, referred to as implicit gradients in this paper. A problem-independent shock capturing approach via Boundary Variation Diminishing (BVD) algorithm is used to suppress oscillations for the simulation of flows with shocks and material interfaces. Several numerical test cases are carried out to verify the proposed method’s capability using the implicit gradient method for compressible flows.

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.

A new high-order compact and conservative numerical scheme for the generalized symmetric regularized long wave equations

ABSTRACT In this paper, a new three-level implicit and fourth-order compact conservative difference scheme is developed for the generalized symmetric regularized long wave equations. The proposed scheme adopts a new time discretization method. The conservation of the new compact scheme is proved, and the discrete conservation laws are obtained. The priori estimation in the maximum norm is discussed by the discrete energy method and mathematical induction. Based on the priori estimation and Brouwer's fixed point theorem, the unique existence of the difference solution is proved. It is proved that the compact difference scheme is unconditionally convergent and stable in the maximum norm, and its convergence order is the fourth order in space and the second order in time. A decoupled and linearized iterative algorithm is proposed to calculate the nonlinear algebraic system generated by the compact scheme, and its convergence is proved. Several numerical experiments are performed to prove the efficiency and reliability of the proposed numerical scheme and to confirm the results obtained from the theoretical analysis.

Arbitrarily High-Order Conservative Schemes for the Generalized Korteweg--de Vries Equation

This paper proposes a new class of arbitrarily high-order conservative numerical schemes for the generalized Korteweg--de Vries (KdV) equation. This approach is based on the scalar auxiliary variable method. The equation is reformulated into an equivalent system by introducing a scalar auxiliary variable, and the energy is reformulated into a sum of two quadratic terms. Therefore, the quadratic preserving Runge--Kutta method will preserve all the three invariants (momentum, mass, and the reformulated energy) in the discrete time flow (assuming the spatial variable is continuous). With the Fourier pseudospectral spatial discretization, the scheme conserves the first and third invariant quantities (momentum and energy) exactly in the space-time full discrete sense. The discrete mass possesses the precision of the spectral accuracy. Our numerical experiment shows the great efficiency of this scheme in simulating the breathers for the modified KdV equation.

A Wave-Targeted Essentially Non-Oscillatory 3D Shock-Capturing Scheme for Breaking Wave Simulation

A new three-dimensional high-order shock-capturing model for the numerical simulation of breaking waves is proposed. The proposed model is based on an integral contravariant form of the Navier–Stokes equations in a time-dependent generalized curvilinear coordinate system. Such an integral contravariant form of the equations of motion is numerically integrated by a new conservative numerical scheme that is based on three elements of originality: the time evolution of the state of the system is carried out using a predictor–corrector method in which exclusively the conserved variables are used; the point values of the conserved variables on the cell face of the computational grid are obtained using an original high-order reconstruction procedure called a wave-targeted essentially non-oscillatory scheme; the time evolution of the discontinuity on the cell faces is calculated using an exact Riemann solver. The proposed model is validated by numerically reproducing several experimental tests of breaking waves on computational grids that are significantly coarser than those used in the literature to validate the existing 3D shock-capturing models. The results obtained with the proposed model are also compared with those obtained with a previously published model, which is based on second-order total variation diminishing reconstructions and an approximate Riemann solver usually adopted in the existing 3D shock-capturing models. Through the above comparison, the main drawbacks of the existing 3D shock-capturing models and the ability of the proposed model to simulate breaking waves and wave-induced currents are shown. The proposed 3D model is able to correctly simulate the wave height increase in the shoaling zone and to effectively predict the location of the wave breaking point, the maximum wave height, and the wave height decay in the surf zone. The validated model is applied to the simulation of the interaction between breaking waves and an emerged breakwater. The numerical results show that the proposed model is able to simulate both the large-scale circulation patterns downstream of the barrier and the onset of quasi-periodic vortex structures close to the edge of the barrier.

Higher-Order Accurate and Conservative Hybrid Numerical Scheme for Relativistic Time-Fractional Vlasov-Maxwell System

The historical analysis demonstrates that plasma scientists produced a variety of numerical methods for solving “kinetic” models, i.e., the Vlasov-Maxwell (VM) system. Still, on the other hand, a significant fact or drawback of most algorithms is that they do not preserve conservation philosophies. This is a crucial fact that cannot be disregarded since the Vlasov Maxwell system is associated with conservation rules and is capable of assessing after the accomplishment of certain helpful mathematical actions. To examine the fractional-order routine of charged particles, we constructed a fractional-order plasma model and proposed a higher-order conservative numerical approach based on operational matrices theory and Shifted Gegenbauer estimations. Numerical convergence is investigated to confirm its competence and compatibility. This concept may be used in problems involving variable order and multidimensionality, such as those involving Vlasov and Boltzmann systems.

A consistent mass–momentum flux computation method for the simulation of plunging jet

In the present study, a robust and conservative numerical scheme is proposed to simulate the violent two-phase flows with high-density ratios. In this method, the mass conservation equation and the momentum equation are solved in a consistent manner. The tangent of hyperbola for interface capturing scheme is extended for the computation of the mass flux by which the sharpness and conservation property of density field is preserved. Compared with other recently proposed methods, no geometrical computation is involved in deriving the mass flux and the spurious velocity in the interfacial region can be completely avoided. To improve the computational efficiency, the present method is implemented on a parallel block-structured adaptive mesh refinement method with a staggered layout of variables. High-fidelity numerical simulation of plunging jet through the liquid surface is performed. A bubble detection algorithm is developed to track bubbles generated in air entrainment process. The evolution of the bubble cloud, air concentration, bubble-size, and bubble-velocity distributions are predicted and compared quantitatively with the experiment. Numerical results show the air entrainment and penetration depth are highly correlated with the upstream disturbance. The growing interfacial roughness of the jet yields more entrained air in the final stage of jet impingement. It is found that when the initial perturbation is introduced, the overall size of the equivalent bubble radius will expand, and the penetration depth of the bubble cloud will decrease, while a larger volume of air is entrained.

A new conservative finite difference scheme for 1D Cahn–Hilliard equation coupled with elasticity

Abstract In this article, we give analysis for a structure-preserving finite difference scheme to the Cahn–Hilliard system coupled with elasticity in one space dimension. In the previous article [K. Shimura and S. Yoshikawa, Error estimate for structure-preserving finite difference schemes of the one-dimensional Cahn–Hilliard system coupled with viscoelasticity, Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations, RIMS Kôkyûroku Bessatsu B82, Research Institute for Mathematical Sciences (RIMS), Kyoto 2020, 159–175], we studied the system coupled with viscoelasticity, where we proposed a conservative numerical scheme for the system which inherits the total energy conservation and momentum conservation laws, and showed the error estimate. However, the error estimate can not be applied to the system without viscosity, due to the fact that the proof relies on the viscous term. Here, we show the error estimate for the system without viscosity by proposing a new structure-preserving finite difference scheme for the system. In addition, we also give the proof of existence of solution for the scheme.

Разработка явных и консервативных схем для решеточных уравнений Больцмана с адаптивным переносом

The Lattice Boltzmann Method (LBM) has several limitations for velocity and temperature. One can consider distribution function in moving frame to overcome these limitations as in PonD. In PonD, values of distribution functions are streamed from off-lattice points, so value estimation is needed. It leads to the implicit and non-conservative numerical scheme. Earlier, for the one-dimensional case, the approach of moments prediction was found, which leads to an explicit and conservative numerical scheme. We apply this approach to the two-dimensional and three-dimensional cases in this work. Requirements to interpolation stencil, quadrature, and Hermite polynomial expansion which guarantee moment matching, conservation, and exact calculation, were studied. The resulting schemes were implemented and tested on several tasks.

Mathematical analysis of a conservative numerical scheme for the Ostrovsky equation

Mathematical analysis of a conservative numerical scheme for the Ostrovsky equation

Conservative numerical schemes with optimal dispersive wave relations: Part I. Derivation and analysis

Conservative numerical schemes with optimal dispersive wave relations: Part I. Derivation and analysis

Resolving Wave Propagation in Anisotropic Poroelastic Media Using Graphical Processing Units (GPUs)

AbstractBiot's equations describe the physics of hydromechanically coupled systems establishing the widely recognized theory of poroelasticity. This theory has a broad range of applications in Earth and biological sciences as well as in engineering. The numerical solution of Biot's equations is challenging because wave propagation and fluid pressure diffusion processes occur simultaneously but feature very different characteristic time scales. Analogous to geophysical data acquisition, high resolution and three dimensional numerical experiments lately redefined state of the art. Tackling high spatial and temporal resolution requires a high‐performance computing approach. We developed a multi‐ graphical processing units (GPU) numerical application to resolve the anisotropic elastodynamic Biot's equations that relies on a conservative numerical scheme to simulate, in a few seconds, wave fields for spatial domains involving more than 1.5 billion grid cells. We present a comprehensive dimensional analysis reducing the number of material parameters needed for the numerical experiments from ten to four. Furthermore, the dimensional analysis emphasizes the key material parameters governing the physics of wave propagation in poroelastic media. We perform a dispersion analysis as function of dimensionless parameters leading to simple and transparent dispersion relations. We then benchmark our numerical solution against an analytical plane wave solution. Finally, we present several numerical modeling experiments, including a three‐dimensional simulation of fluid injection into a poroelastic medium. We provide the Matlab, symbolic Maple, and GPU CUDA C routines to reproduce the main presented results. The high efficiency of our numerical implementation makes it readily usable to investigate three‐dimensional and high‐resolution scenarios of practical applications.

A consistent and conservative volume distribution algorithm and its applications to multiphase flows using Phase-Field models

In the present study, the multiphase volume distribution problem, where there can be an arbitrary number of phases, is addressed using a consistent and conservative volume distribution algorithm. The proposed algorithm satisfies the summation constraint, the conservation constraint, and the consistency of reduction. The first application of the volume distribution algorithm is to determine the Lagrange multipliers in multiphase Phase-Field models that enforce the mass conservation, and a multiphase conservative Allen-Cahn model that satisfies the consistency of reduction is developed. A corresponding consistent and conservative numerical scheme is developed for the model. The multiphase conservative Allen-Cahn model has a better ability than the multiphase Cahn-Hilliard model to preserve under-resolved structures. The second application is to develop a numerical procedure, called the boundedness mapping, to map the order parameters, obtained numerically from a multiphase model, into their physical interval, and at the same time to preserve the physical properties of the order parameters. Along with the consistent and conservative schemes for the multiphase Phase-Field models, the numerical solutions of the order parameters are reduction consistent, conservative, and bounded, which are theoretically analyzed and numerically validated. Then, the multiphase Phase-Field models are coupled with the momentum equation by satisfying the consistency of mass conservation and the consistency of mass and momentum transport, thanks to the consistent formulation. It is demonstrated that the proposed model and scheme converge to the sharp-interface solution and are capable of capturing the complicated multiphase dynamics even when there is a large density and/or viscosity ratio.

A Lagrangian theory of geostrophic adjustment for zonally invariant flows on a rotating spherical earth

We examine the late-time evolution of an inviscid zonally symmetric shallow-water flow on the surface of a rotating spherical earth. An arbitrary initial condition radiates inertia–gravity waves that disperse across the spherical surface. The simpler problem of a uniformly rotating (f-plane) shallow-water flow on the plane radiates these waves to infinity, leaving behind a nontrivial steady flow in geostrophic balance (in which the Coriolis acceleration balances the horizontal hydrostatic pressure gradient). This is called “geostrophic adjustment.” On a sphere, the waves cannot propagate to infinity, and the flow can never become steady due to energy conservation (at least in the absence of shocks). Nonetheless, when energy is conserved a form of adjustment still takes place, in a time-averaged sense, and this flow satisfies an extended form of geostrophic balance dependent only on the conserved mass and angular momentum distributions of fluid particles, just as in the planar case. This study employs a conservative numerical scheme based on a Lagrangian form of the rotating shallow-water equations to substantiate the applicability of these general considerations on an idealized aqua-planet for an initial “dam” along the equator in a motionless ocean.

A conservative numerical scheme for capturing interactions of optical solitons in a 2D coupled nonlinear Schrödinger system

In this study, an efficient fourth-order conservative explicit numerical scheme using method of lines is developed to simulate different scenarios of soliton interactions and reflections for a (2 + 1)-dimensional coupled nonlinear Schrodinger (CNLS) system. The fourth-order Runge–Kutta technique is applied as a time integrator to the resulting ordinary differential system. Both integrable and nonintegrable cases of the CNLS system are considered. A condition for the scheme to be stable is deduced with the aid of von Neumann stability analysis. Several numerical experiments have been carried out to exhibit the reliability of the scheme in capturing and understanding the interesting phenomenon of elastic and inelastic soliton collisions/reflections related to many nonlinear evolution equations. The ability of the scheme to preserve the conserved invariants in long terms confirms its accuracy and stability. New results associated with interactions and reflections of soliton waves are obtained.

Quantum Fokker-Planck modeling of degenerate electrons

An implicit and conservative numerical scheme is proposed for the isotropic quantum Fokker-Planck equation describing the evolution of degenerate electrons subject to elastic collisions with other electrons and ions. The electron-ion and electron-electron collision operators are discretized using a discontinuous Galerkin method, and the electron energy distribution is updated by an implicit time integration method. The numerical scheme is designed to satisfy all conservation laws exactly. Numerical tests and comparisons with other modeling approaches are shown to demonstrate the accuracy and conservation properties of the proposed method.

Conservative Numerical Schemes for the Nonlinear Fractional Schrödinger Equation

Conservative Numerical Schemes for the Nonlinear Fractional Schrödinger Equation

Numerical Method for Solving a System of Kinetic Equations Describing the Behavior of a Nonideal Gas

A previously constructed kinetic model for describing the behavior of a nonideal gas is investigated. The dimensionless parameters determining when the nonideal nature of the gas has to be taken into account are estimated in more detail. It is found that the collision integral for bound particles can be integrated over the velocity space, which significantly simplifies the original system of equations and makes it possible to prove an H-theorem. The resulting system is nondimensionalized. A conservative numerical scheme is proposed for its solution.