Semi-implicit Finite Difference Scheme Research Articles

Chaotic Communication Systems with Signal Modulation Based on Controlled Symmetry of Semi-Implicit Finite-Difference Models

The article is devoted to investigation coherent communication system model with a new method of signal modulation based on variable symmetry of finite-difference schemes with subsequent experimental analysis of the effectiveness of different modulation techniques. The aim of the study is to investigate a computer model of chaotic communication system with signal modulation based on variable symmetry of semi-implicit finite-difference schemes. Novelty: elements of scientific novelty have finite-difference models of receivers/transmitters, allowing to realize a new method of modulation of chaotic signals. Result: obtaining a simulation model of coherent chaotic communication systems with tools for covertness and noise immunity analyses. Practical relevance: The simulation model of chaotic communication system is a necessary tool for analyzing the performance of the system before its physical implementation.

Unconditionally stable higher order semi-implicit level set method for advection equations

We present compact semi-implicit finite difference schemes on structured grids for numerical solutions of advection by an external velocity and by a speed in the normal direction that are applicable in level set methods. The recommended numerical scheme is third order accurate for the linear advection in the 2D case with a space dependent velocity. Using a combination of analytical and numerical tools in the von Neumann stability analysis, the third order scheme is claimed to be unconditionally stable. We also present a simple high-resolution scheme that gives a TVD (Total Variation Diminishing) approximation of the spatial derivative for the advected level set function in the 1D case. In the case of nonlinear advection, a semi-implicit discretization is proposed to linearize the problem. The compact implicit part of the stencil of numerical schemes contains unknowns only in the upwind direction. Consequently, algebraic solvers like the fast sweeping method can be applied efficiently to solve the resulting algebraic systems. Numerical tests to evolve a smooth and non-smooth interface and an example with a large variation of the velocity confirm the good accuracy of the third order scheme even in the case of very large Courant numbers. The advantage of the high-resolution scheme is documented for examples where the advected level set functions contain large jumps in the gradient.

The splitting-based semi-implicit finite-difference schemes for simulation of blood flow in arteries

This paper is devoted to construction and analysis of splitting-based finite-difference schemes for simulation of blood flow in applications of one-dimensional models. Proposed schemes are constructed for inviscid and viscid models of blood. The proposed approach is based on the modification of averaged incompressibility condition, rewritten for the square root of a cross-sectional area. It is approximated by an absolutely stable scheme with a skew-symmetric operator. The constructed schemes can be classified as semi-implicit. The stability of proposed schemes is analyzed by the method of energy inequalities. Sufficient stability conditions are obtained. For the problems with analytical solutions, it is demonstrated that a second-order convergence rate is achieved in practice.The proposed schemes are compared with the widely used explicit finite-difference schemes, such as Lax–Wendroff, Lax–Friedrichs and McCormack schemes. The benchmark problems for simulation of flows in the following vascular systems are considered: human carotid artery, human aorta with bifurcation, vessel with bifurcation and absorbing outflow conditions, and networks with 31, 63, and 127 vessels. It is demonstrated that the computations based on the splitting-based semi-implicit schemes can be performed much faster than for the explicit schemes with close values of relative errors.

High resolution finite difference schemes for a size structured coagulation-fragmentation model in the space of radon measures.

In this paper, we develop explicit and semi-implicit second-order high-resolution finite difference schemes for a structured coagulation-fragmentation model formulated on the space of Radon measures. We prove the convergence of each of the two schemes to the unique weak solution of the model. We perform numerical simulations to demonstrate that the second order accuracy in the Bounded-Lipschitz norm is achieved by both schemes.

Positivity-preserving and energy-dissipative finite difference schemes for the Fokker–Planck and Keller–Segel equations

Abstract In this work we introduce semi-implicit or implicit finite difference schemes for the continuity equation with a gradient flow structure. Examples of such equations include the linear Fokker–Planck equation and the Keller–Segel equations. The two proposed schemes are first-order accurate in time, explicitly solvable, and second-order and fourth-order accurate in space, which are obtained via finite difference implementation of the classical continuous finite element method. The fully discrete schemes are proved to be positivity preserving and energy dissipative: the second-order scheme can achieve so unconditionally while the fourth-order scheme only requires a mild time step and mesh size constraint. In particular, the fourth-order scheme is the first high order spatial discretization that can achieve both positivity and energy decay properties, which is suitable for long time simulation and to obtain accurate steady state solutions.

Numerical investigation of the onset of three-dimensional characteristics in flow past a pair of square cylinders at various arrangements

An incompressible Navier–Stokes solver has been developed to analyze the three-dimensional viscous flow around a pair of square cylinders at a low Reynolds number (Re = 100) at various arrangements. A semi-implicit finite difference scheme was employed on a staggered grid, which facilitated the easier application of boundary conditions while achieving satisfactory accuracy and stability. A third-order upwind (Kawamura–Kuwahara) scheme was applied to discretize the convective terms and a second-order central difference scheme to discretize the diffusive terms. Time integration was performed using a second-order accurate Adams–Bashforth scheme. An iterative pressure–velocity correction was used to obtain a divergence-free velocity field. Aspects of the flow such as vorticity profiles, the time evolution of lift and drag coefficients, and three-dimensionality appearance were studied. Three-dimensional characteristics ceased to appear in a tandem arrangement, irrespective of the gap ratio. In contrast, the side by side arrangement showed a clear three-dimensional transition in the flow when analyzing the vortex structures. Further investigations carried out for staggered arrangement keeping the center to center distance constant and varying the stagger angle (α) reveal that the three-dimensional behavior starts at α = 30° and keeps progressing with the angle increase.

Convergence of a higher order finite difference scheme for two-phase Stefan problems.

This work establishes a sufficient condition for convergence of a semi-implicit finite difference scheme with variable space grid of fourth order of accuracy in space, previously set by the authors. The scheme is intended to solve one-dimensional two-phase Stefan problems. The proofs of the above-mentioned results are based on estimates which are obtained and subsequently employed in order to find an upper bound of the discretization error both at phases and the interface. The computational behavior of the scheme under the sufficient condition is corroborated by a series of numerical tests.

Simulation of activator–inhibitor dynamics based on cross-diffusion Brusselator reaction–diffusion system via a differential quadrature-radial point interpolation method (DQ-RPIM) technique

The current paper proposes an efficient numerical procedure for solving the two-dimensional Brusselator reaction–diffusion system. First, the time derivative is discretized by a semi-implicit finite difference scheme such that the nonlinear terms are approximated by using one term of the Taylor expansion. It is shown if the nonlinear terms satisfy the Lipshitz condition, then the proposed difference scheme is stable. In the next, the spatial derivative is discretized by the differential quadrature technique based upon the shape functions of radial point interpolation method. We study some practical problems to show the efficiency and acceptable results of the developed technique.

Optimal Error Analysis of Euler and Crank--Nicolson Projection Finite Difference Schemes for Landau--Lifshitz Equation

The Landau--Lifshitz equation has been widely used to describe the dynamics of magnetization in a ferromagnetic material, which is highly nonlinear with the nonconvex constraint $|{m}|=1$. A crucial issue in designing efficient numerical schemes is to preserve this constraint in the discrete level. A simple and frequently used one is the projection method, which projects the numerical solution onto a unit sphere at each time step. The method has been used in many areas in the past several decades, while analysis has not been explored. In this paper, we present optimal error analysis of a backward Euler and a Crank--Nicolson semi-implicit projection finite difference scheme for the Landau--Lifshitz equation. The analysis is based on new and precise estimates of the difference between the errors of projected and unprojected solutions in both $L^2$ and $H^1$ norms. Some numerical experiments are provided to confirm our theoretical results.

Two Dimension Vertically Integrated Numerical Suspended Sediment Transport Model (Dept.C)

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A semi-implicit finite difference scheme for the multi-term time-fractional Burgers-type equations

The main purpose of this paper is to construct a semi-implicit difference scheme for the multi-term time-fractional Burgers-type equations. Firstly, the L2-discretization formula is applied to the discretization of the multi-term Caputo fractional derivatives. Secondly, the second-order spatial derivative is approximated by using the second-order central difference quotient approximation and the nonlinear convection term $$uu_x$$ is discretized via the semi-implicit method. Then, a fully discrete finite difference scheme is established. The unconditional stability and convergence in maximum-norm are derived by the discrete energy method and the mathematical induction. Numerical experiments are performed to validate the theoretical analysis.

A Linearized Finite Difference Scheme for the Richards Equation Under Variable-Flux Boundary Conditions

The Richards equation is a degenerate nonlinear PDE that models a flow through saturated/unsaturated porous media. Research on its numerical methods has been conducted in many fields. Implicit schemes based on a backward Euler format are widely used in calculating it. However, it is difficult to obtain stability with a numerical scheme because of the strong nonlinearity and degeneracy. In this paper, we establish a linearized semi-implicit finite difference scheme that is faster than backward Euler implicit schemes. We analyze the stability of this scheme by adding a small positive perturbation $$\epsilon $$ to the coefficient function of the Richards equation. Moreover, we show that there is a linear relationship between the discretization error in the $$L^{\infty }$$-norm and $$\epsilon $$. Numerical experiments are carried out to verify our main results.

A fully discrete positivity-preserving and energy-dissipative finite difference scheme for Poisson–Nernst–Planck equations

The Poisson–Nernst–Planck (PNP) equations is a macroscopic model widely used to describe the dynamics of ion transport in ion channels. In this paper, we introduce a semi-implicit finite difference scheme for the PNP equations in a bounded domain. A general boundary condition for the Poisson equation is considered. The fully discrete scheme is shown to satisfy the following properties: mass conservation, unconditional positivity, and energy dissipation (hence preserves the steady state). Solvability of the semi-discrete scheme is proved and a simple fixed point iteration is proposed to solve the fully discrete scheme. Numerical examples in both 1D and 2D and for multiple species are presented to demonstrate the convergence and properties of the proposed scheme.

Efficient semi-implicit compact finite difference scheme for nonlinear Schrödinger equations on unbounded domain

The compact finite difference scheme is designed to numerically solve the nonlinear Schrödinger equation and coupled nonlinear Schrödinger equations on an unbounded domain in this paper. The original problem on an unbounded domain is reduced to an initial boundary value problem defined on a bounded computational domain by applying the artificial boundary method. Then, the reduced problem on the bounded computational domain is solved by an efficient semi-implicit compact finite difference scheme, which is a fourth-order scheme with respect to spatial variable. The scheme efficiently avoids the time-consuming iteration procedure necessary for the nonlinear scheme and thus is relatively time-saving. Finally, the stability of the proposed scheme is rigorously analyzed. Numerical examples are given to illustrate the accuracy and effectiveness of the proposed method.

A second order unconditionally stable scheme for the modified phase field crystal model with elastic interaction and stochastic noise effect

In this paper, we extend the phase field crystal model to the modified phase field crystal model which includes diffusive dynamics, elastic interaction, and stochastic noises effect. We present a second-order accurate semi-implicit finite difference scheme for the modified phase field crystal model. The resulting scheme is based on the stabilized splitting method and Crank–Nicolson method. The nonlinear term is linearized by the Taylor series. The resulting scheme is linear at each time step, which makes it easy to be implemented and efficient to be solved by using the linear multigrid solver. We prove that the resulting scheme is unconditionally energy stable. Various numerical experiments are conducted to verify the accuracy and efficiency of our proposed algorithm.

Three semi-implicit compact finite difference schemes for the nonlinear partial integro-differential equation arising from viscoelasticity

ABSTRACT Three semi-implicit compact finite difference schemes are described for the nonlinear partial integro-differential equation arising from viscoelasticity. In our methods, the time derivative is approximated by the backward-Euler, Crank–Nicolson-type and formally second-order backward differentiation formula scheme, respectively, and the convolution quadrature formula is employed to process the Riemann–Liouville fractional integral term. Fully discrete difference schemes are constructed with the spatial discretization by the compact finite difference formula. Meanwhile, the semi-implicit technique is used to deal with nonlinear convection term uux . In the numerical experiment, the comparisons between our methods and existing methods demonstrate the efficiency of our methods.

A simple and efficient numerical model for simulating one dimensional dam break flows

A semi implicit finite difference scheme is presented in this paper for simulation of dam break flow over undulated bed. For this purpose, the one dimensional (1D) governing equations of shallow water flow, in a deviatoric version is solved here. One of the main important features of the present model is that, no special treatment is required here to deal with the offending friction slope source term for very less water depth. Regarding the bed slope term also, simple central difference approximation is sufficient to handle undulated channel bed. To investigate the performance of the present model in diverse situations, it is used to replicate different dam break test problems including one of known analytical solution and two experimental dam break cases. A detail analysis of the results show that the model capable is of producing good quality results for varied conditions. A sensitivity analysis is carried out for the damping parameter and an optimal range is proposed for it.

Unconditionally Stable One-Step Leapfrog ADI-FDTD for Dispersive Media

A novel unconditionally stable one-step leapfrog alternating-direction-implicit finite-difference time domain is developed for modeling dispersive media. In contrast to previous approaches, the proposed method uses a vector fitting technique to incorporate various types of dispersive media through electric polarization terms governed by an auxiliary differential equation (ADE). Moreover, a semi-implicit finite-difference scheme is applied to the ADE to maintain the unconditional stability of the method. The stability is verified analytically by von Neumann analysis with the Jury criterion. Numerical experiments are carried out to illustrate the stability and accuracy. The proposed method is used to investigate surface plasmon polaritons (SPPs) on graphene sheets biased by an electrostatic field.

Parallel energy-stable phase field crystal simulations based on domain decomposition methods

In this paper, we present a parallel numerical algorithm for solving the phase field crystal equation. In the algorithm, a semi-implicit finite difference scheme is derived based on the discrete variational derivative method. Theoretical analysis is provided to show that the scheme is unconditionally energy stable and can achieve second-order accuracy in both space and time. An adaptive time step strategy is adopted such that the time step size can be flexibly controlled based on the dynamical evolution of the problem. At each time step, a nonlinear algebraic system is constructed from the discretization of the phase field crystal equation and solved by a domain decomposition based, parallel Newton–Krylov–Schwarz method with improved boundary conditions for subdomain problems. Numerical experiments with several two and three dimensional test cases show that the proposed algorithm is second-order accurate in both space and time, energy stable with large time steps, and highly scalable to over ten thousands processor cores on the Sunway TaihuLight supercomputer.

CIP Method of Characteristics for the Solution of Tide Wave Equations

The CIP-MOC (Constrained Interpolation Profile/Method of Characteristics) is proposed to solve the tide wave equations with large time step size. The bottom topography and bottom friction, which are very important factor for the tidal wave model, are included to the equation of Riemann invariants as the source term. Numerical experiments demonstrate the good performance of the scheme. Compared to traditional semi-implicit (SI) finite difference scheme which is widely used in tidal wave simulation, CIP-MOC has better stability in simulating large gradient water surface change and has the ability to use much longer time step size under the premise of maintaining accuracy. Besides, numerical tests with reflective boundary conditions are carried out by CIP-MOC with large time step size and good results are obtained.