Conservative Finite Difference Scheme Research Articles

Unconditionally Stable Algorithm for Solving the Three-Dimensional Nonstationary Navier–Stokes Equations

We propose an unconditionally stable method for solving the three-dimensional nonstationary Navier–Stokes equations in the velocity–pressure variables. The method is based on a conservative finite-difference scheme and the simultaneous solution of the momentum and continuity equations at each time layer. The velocity and pressure fields are calculated by using a parallel algorithm for solving systems of linear equations by the Gauss method.

An improved consistent, conservative, non-oscillatory and high order finite difference scheme for variable density low Mach number turbulent flow simulation

A consistent, conservative and truly high order finite difference scheme for low Mach number turbulent variable density flow simulation is presented. In the present scheme, the scalar transport equation is discretized with the explicit weighted compact nonlinear scheme (WCNS) while the continuity and momentum equations are discretized with a conservative centered high order finite difference scheme on a staggered grid. Bounded hybrid high order upwind interpolation is proposed for the evaluation of density on the cell faces of the staggered grid. The use of explicit WCNS along with the bounded density interpolation resolves the issue of degraded order of accuracy in the existing high order finite difference schemes where the centered finite difference scheme for mass and momentum conservation is combined with a total variation stable scheme for scalar transport. With the same difference operator used for the continuity and scalar transport equations, the present scheme is consistent naturally, and resolves the inconsistency problem reported in Trisjono et al. (2016) [6] without employing any modification to a scheme. Such improvements are demonstrated for several test cases.

Numerical method and simplified analytical model for predicting the blast load in a partially confined chamber

The paper presents a study aimed at understanding the characteristics of an internal explosion within a chamber with limited venting. The study includes numerical simulations and analytical derivations. An in-house 3D code employing an improved weighted essentially non-oscillatory (WENO) conservative finite difference scheme was used to carry out the simulations. It is indicated that the proposed improved WENO scheme can resolve the shock waves with higher accuracy and resolution. Further, a simplified analytical model to predict the quasi-static overpressure was developed based on the conservation law of total energy and dimensional analysis theory. It is demonstrated that the proposed simplified approach for prediction of the quasi-static overpressure agrees well with simulation results for a wide range of explosive weights and venting hole sizes. The studies in this paper provide an efficient method to predict the blast load inside a partially confined chamber for the analysis of the consequences of explosion.

Hybrid Compact-WENO Finite Difference Scheme with Radial Basis Function Based Shock Detection Method for Hyperbolic Conservation Laws

A hybrid scheme, based on the high order nonlinear characteristicwise weighted essentially nonoscillatory (WENO) conservative finite difference scheme and the spectral-like linear compact finite difference scheme, has been developed for capturing shocks and strong gradients accurately and resolving fine scale structures efficiently for hyperbolic conservation laws. The key issue in any hybrid scheme is the design of an accurate, robust, and efficient high order shock detection algorithm which is capable of determining the smoothness of the solution at any given grid point and time. An improved iterative adaptive multiquadric radial basis function (IAMQ-RBF-Fast) method [W. S. Don, B. S. Wang, and Z. Gao, J. Sci. Comput, 75 (2018), pp. 1016--1039], which employed the $O(N^2)$ recursive Levinson--Durbin method and the Sherman--Morrison--Woodbury method for solving the perturbed Toeplitz matrix system, has been successfully developed as an efficient and accurate edge detector of the piecewise smooth functions. In this study, the method, together with Tukey's boxplot method and the domain segmentation technique, is extended to serve as a novel shock detection algorithm for solving the Euler equations. The applicability and performance of the RBF edge detection method as the shock detector in the hybrid scheme in terms of accuracy, robustness, efficiency, resolution, and other implementation issues are given. Several one- and two-dimensional benchmark problems in shocked flow demonstrate that the proposed hybrid scheme can reach a speedup of the CPU times by a factor up to 2--3 compared with the pure fifth order WENO-Z scheme.

Conservative Finite-Difference Scheme and Two-Stage Iteration Process of Its Realization for the 2D Problem of Semiconductor Plasma Generation by Femtosecond Pulse

Conservative Finite-Difference Scheme and Two-Stage Iteration Process of Its Realization for the 2D Problem of Semiconductor Plasma Generation by Femtosecond Pulse

A Conservative Difference Scheme for Space Fractional Klein-Gordon-Schrödinger Equations with a High-Degree Yukawa Interaction

Abstract A conservative finite difference scheme for nonlinear space fractional Klein-Gordon-Schrödinger systems with high-degree Yukawa interaction is studied. We show that the arising difference equations are uniquely solvable and approximate solutions converge to the exact solution at the rate O ($τ^2+h^2$). Moreover, we prove that the scheme can be decoupled and preserves the mass and energy conservation laws. Numerous examples confirm theoretical results and demonstrate the efficiency of the scheme. They also show the influence of the fractional order and the high-degree term coefficient on the shape and the propagation velocity of solitary waves.

An efficient and conservative compact finite difference scheme for the coupled Gross–Pitaevskii equations describing spin-1 Bose–Einstein condensate

The coupled Gross–Pitaevskii system studied in this paper is an important mathematical model describing spin-1 Bose-Einstein condensate. We propose a linearized and decoupled compact finite difference scheme for the coupled Gross–Pitaevskii system, which means that only three tri-diagonal systems of linear algebraic equations at each time step need to be solved by using Thomas algorithm. New types of mass functional, magnetization functional and energy functional are defined by using a recursive relation to prove that the new scheme preserves the total mass, energy and magnetization in the discrete sense. Besides the standard energy method, we introduce an induction argument as well as a lifting technique to establish the optimal error estimate of the numerical solution without imposing any constraints on the grid ratios. The convergence order of the new scheme is of O(h4+τ2) in the L2 norm and H1 norm, respectively, with time step τ and mesh size h. Our analysis method can be used to high dimensional cases and other linearized finite difference schemes for the two- or three-dimensional nonlinear Schrödinger/Gross–Pitaevskii equations. Finally, numerical results are reported to test the theoretical results.

Evolution to a steady state for a rarefied gas flowing from a tank into a vacuum through a plane channel

A kinetic equation (S-model) is used to solve the nonstationary problem of a monatomic rarefied gas flowing from a tank of infinite capacity into a vacuum through a long plane channel. Initially, the gas is at rest and is separated from the vacuum by a barrier. The temperature of the channel walls is kept constant. The flow is found to evolve to a steady state. The time required for reaching a steady state is examined depending on the channel length and the degree of gas rarefaction. The kinetic equation is solved numerically by applying a conservative explicit finite-difference scheme that is firstorder accurate in time and second-order accurate in space. An approximate law is proposed for the asymptotic behavior of the solution at long times when the evolution to a steady state becomes a diffusion process.

A three-level linear implicit conservative scheme for the Rosenau–KdV–RLW equation

A three-level linear conservative implicit finite difference scheme for solving the Rosenau–KdV–RLW equation is proposed. The existence, uniqueness, and conservation for energy of the difference solution are proved by the discrete energy method. The new scheme is 2nd-order convergent and unconditionally stable. Several numerical experiments verify that the theoretical results are accurate and the scheme is efficient and reliable.

Unconditional and optimal H 2-error estimates of two linear and conservative finite difference schemes for the Klein-Gordon-Schrödinger equation in high dimensions

The focus of this paper is on the optimal error bounds of two finite difference schemes for solving the d-dimensional (d = 2, 3) nonlinear Klein-Gordon-Schrodinger (KGS) equations. The proposed finite difference schemes not only conserve the mass and energy in the discrete level but also are efficient in practical computation because only two linear systems need to be solved at each time step. Besides the standard energy method, an induction argument as well as a ‘lifting’ technique are introduced to establish rigorously the optimal H 2-error estimates without any restrictions on the grid ratios, while the previous works either are not rigorous enough or often require certain restriction on the grid ratios. The convergence rates of the proposed schemes are proved to be at O(h 2 + τ 2) with mesh-size h and time step τ in the discrete H 2-norm. The analysis method can be directly extended to other linear finite difference schemes for solving the KGS equations in high dimensions. Numerical results are reported to confirm the theoretical analysis for the proposed finite difference schemes.

Correlation between elastic energy density and deep earthquakes distribution

The mechanism at the origin of the earthquakes below 30km remains elusive as these events cannot be explained by brittle frictional processes. In this work we focus on the global total distribution of earthquakes frequency vs. depth from ∼50km to 670km depth. We develop a numerical model of self-driven subduction by solving the non-homogeneous Stokes equation using the “Particle in cell method” in combination with a conservative finite difference scheme, here solved for the first time using Python and NumPy only. We show that most of the elastic energy is stored in the slab core and that it is strongly correlated with the earthquake frequency-depth distribution for a wide range of lithosphere and lithosphere-core viscosities. According to our results, we suggest that 1) slab bending at the bottom of the upper mantle causes the peak of the earthquake frequency-depth distribution that is observed at mantle transition depth; 2) the presence of a high viscous stiff core inside the lithosphere generates an elastic energy distribution that fits better with the exponential decay that is observed at intermediate depth.

On the Conservative Finite Difference Scheme for the Generalized Novikov Equation

In this paper, we investigate a numerical method for the generalized Novikov equation. We propose a conservative finite difference scheme and use Brouwer fixed point theorem to obtain the existence of the solution of the corresponding difference equation. We also prove the convergence and stability of the solution by using the discrete energy method. Moreover, we obtain the truncation error of the difference scheme which is .

Conservative Methods for Dynamical Systems

We show a novel systematic way to construct conservative finite difference schemes for quasilinear first-order system of ordinary differential equations with conserved quantities. In particular, this includes both autonomous and non-autonomous dynamical systems with conserved quantities of arbitrary forms, such as time-dependent conserved quantities. Sufficient conditions to construct conservative schemes of arbitrary order are derived using the multiplier method. General formulas for first-order conservative schemes are constructed using divided difference calculus. New conservative schemes are found for various dynamical systems such as Euler's equation of rigid body rotation, Lotka-Volterra systems, the planar restricted three-body problem and the damped harmonic oscillator.

A multigrid solution for the Cahn–Hilliard equation on nonuniform grids

We present a nonlinear multigrid method to solve the Cahn–Hilliard (CH) equation on nonuniform grids. The CH equation was originally proposed as a mathematical model to describe phase separation phenomena after the quenching of binary alloys. The model has the characteristics of thin diffusive interfaces. To resolve the sharp interfacial transition, we need a very fine grid, which is computationally expensive. To reduce the cost, we can use a fine grid around the interfacial transition region and a relatively coarser grid in the bulk region. The CH equation is discretized by a conservative finite difference scheme in space and an unconditionally gradient stable type scheme in time. We use a conservative restriction in the nonlinear multigrid method to conserve the total mass in the coarser grid levels. Various numerical results on one-, two-, and three-dimensional spaces are presented to demonstrate the accuracy and effectiveness of the nonuniform grids for the CH equation.

A semi-Lagrangian finite difference WENO scheme for scalar nonlinear conservation laws

For a nonlinear scalar conservation law in one-space dimension, we develop a locally conservative semi-Lagrangian finite difference scheme based on weighted essentially non-oscillatory reconstructions (SL-WENO). This scheme has the advantages of both WENO and semi-Lagrangian schemes. It is a locally mass conservative finite difference scheme, it is formally high-order accurate in space, it has small time truncation error, and it is essentially non-oscillatory. The scheme is nearly free of a CFL time step stability restriction for linear problems, and it has a relaxed CFL condition for nonlinear problems. The scheme can be considered as an extension of the SL-WENO scheme of Qiu and Shu (2011) [2] developed for linear problems. The new scheme is based on a standard sliding average formulation with the flux function defined using WENO reconstructions of (semi-Lagrangian) characteristic tracings of grid points. To handle nonlinear problems, we use an approximate, locally frozen trace velocity and a flux correction step. A special two-stage WENO reconstruction procedure is developed that is biased to the upstream direction. A Strang splitting algorithm is used for higher-dimensional problems. Numerical results are provided to illustrate the performance of the scheme and verify its formal accuracy. Included are applications to the Vlasov–Poisson and guiding-center models of plasma flow.

Numerical simulation of flame acceleration and deflagration-to-detonation transition of ethylene in channels

Ethylene (C2H4) is a hydrocarbon fuel and widely used in chemical industry, however, ethylene is highly flammable and therefore presents a serious fire and explosion hazard. This work is initiated by addressing the hazard assessment of ethylene mixtures in different scale channels (d = 5 mm, 10 mm and 20 mm) from the aspect of flame acceleration (FA) and deflagration-to-detonation transition (DDT) by using large eddy simulation (LES) method coupled with the artificially thickened flame (ATF) approach. The fifth order local characteristics based weighted essentially non-oscillatory (WENO) conservative finite difference scheme is employed to solve the governing equations. The numerical results confirm that flame velocity increase rapidly at the beginning stage in three channels, and the flame acceleration rate is slower in the subsequent stage, afterwards, the flame velocity has an abrupt increase, and the onset of detonation occurs. Due to the fact that wall effect is significant in the narrow channel (e.g.,5 mm), especially in the ignition stage of the flame, flames have different shapes in wider channels (10 mm and 20 mm) and narrow channel (5 mm). Both the pressure and temperature profiles confirm DDT run-up distances are 0.251 m, 0.203 m and 0.161 m in 20 mm, 10 mm and 5 mm channels, respectively, which indicates that a shorter run-up distance is required in narrower channel. The cellular detonation structures for the ethylene-air mixture in different channels indicate that multi-headed detonation structures can be found in 20 mm channel, as the channel width decreases to 10 mm, detonation has a single-headed spinning structure, as the width is further reduced to 5 mm, only large longitudinal oscillation of the pressure can be observed.

Conservative compact finite difference scheme for the coupled Schrödinger–Boussinesq equation

In this article, two conserved compact finite difference schemes for solving the nonlinear coupled Schrödinger–Boussinesq equation are proposed. The conservative property, existence, convergence, and stability of the difference solutions are theoretically analyzed. The numerical results are reported to demonstrate the accuracy and efficiency of the methods and to confirm our theoretical analysis. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1667–1688, 2016

Three-dimensional detonation simulations with the mapped WENO-Z finite difference scheme

Abstract We perform a very long time numerical simulation to capture the three-dimensional detonation structures in a rectangular duct by solving the reactive Euler equations using the high order/resolution WENO-Z conservative finite difference scheme (Gao et al. J. Sci. Comput. 55 : 351–371, 2013). In the algorithm, the perfectly matched layer (PML) absorbing boundary condition (ABC) for the reactive Euler equations is used to reduce the spurious wave reflection from the open left boundary which allows one to use a significant smaller truncated physical domain. Moreover, a tangent grid mapping is used to enhance the grid resolution within the half reaction zone that greatly reduces the memory usage and computational time compared with solving the problem with a uniform grid. The initial Zeldovich-von Neumann-Doring (ZND) profile of two classical stable and slightly unstable detonation waves is perturbed to generate the rectangular in-phase, diagonal in-phase and spinning detonation structures. Depending on the initial perturbation, the stable case shows the presence of a rectangular mode and a diagonal mode of the detonation front, which are suggested to be geometrically similar. The slightly unstable case, as expected, generates the spinning detonations instead in a narrow duct. The results show that a short time simulation is insufficient to capture the cellular detonation structures. The width-to-length ratio of the cellular patterns depends on the gas properties only, but independent of the perturbation of the initial conditions.

Hybrid Compact-WENO Finite Difference Scheme with Conjugate Fourier Shock Detection Algorithm for Hyperbolic Conservation Laws

For discontinuous solutions of hyperbolic conservation laws, a hybrid scheme, based on the high order nonlinear characteristicwise weighted essentially nonoscillatory (WENO) conservative finite difference scheme and the high resolution spectral-like linear compact finite difference (Compact) scheme, is developed for capturing shocks and strong gradients accurately and resolving smooth scale structures efficiently. The key issue in any hybrid scheme is the design of an accurate, robust, and efficient high order shock detection algorithm that is capable of determining the smoothness of the solution at any given grid point. The conjugate Fourier (cF) partial sum and its derivative are investigated for its applicability as a shock detector due to its unique property, namely, the cF partial sum converges to the location and strength of an isolated jump. For a nonperiodic problem, the data are first evenly extended before the derivative of the cF partial sum and its mean are computed. The mean allows one to par...

Geometric multigrid method for steady Buoyancy convection in vertical cylinders

An implementation of the Geometric Multi-Grid (GMG) method, with full approximation scheme/storage (FAS) algorithm, in a numerical study of steady Buoyancy-driven convection with axis-symmetric flows in vertical cylinders is presented. The particular system examined is a cylinder with an aspect ratio a (radius divided by height) of 4. The fluid is heated from below and cooled from above, and the circular wall of the vessel is insulated. The Rayleigh (Ra) and fluid Prandtl (Pr) numbers are respectively 5 6.4 10  and 7. A non-linear system of equations was formulated in stream function-vorticity-temperature variables and discretized using a monotonic conservative finite difference scheme of second order accuracy. The steady state condition was solved for purposes of comparing two numerical methods: the GMG FAS and the GaussSeidel method with lexicographic ordering (GS-LEX). The GMG FAS method has significantly higher efficiency in CPU performance compared to the pure GS-LEX method for fine grids only if the tolerance value for stopping iteration process is chosen not too small. A procedure for the selection of adjustment parameters for GMG FAS algorithm is proposed and tested for different grid sizes. Details regarding convergence criteria are addressed.