Linearized Implicit Scheme Research Articles

Toward robust linear implicit schemes for steady state hypersonic flows

Implicit time discretization in computational fluid dynamics dedicated to compute steady state solution of hypersonic flows was an intense field of research in the 70's-80's. It is suitable for computational efficiency to use implicit schemes that do not suffer from time step restriction to guarantee stability, unlike explicit ones. Unfortunately time step restriction is still required in practice, especially for stiff numerical test cases such as high Mach number flows around objects. A method introduced by Yee et al. (1985) [34] is commonly used to simulate computational fluid dynamics problems in an implicit fashion. However this method has no formal theoretical basis for systems of conservation laws. Consequently the practical time step is driven by a ad-hoc user-given profile. The purpose of this work is first to study the mathematical properties of such linearized implicit finite volume schemes to enlighten their weaknesses and exhibit more adequate linearization processes. We rely on the hyperbolicity of the Euler equations to establish a general framework to design implicit schemes. Secondly, we propose a correction of the system matrix to adapt to any given finite volume scheme. The obtained linearized implicit finite volume methods are more robust and less constrained with regards to ad-hoc user-given profile. Numerical results in 1D and 2D will provide evidences to confirm the analysis on relevant and challenging hypersonic test cases.

A second-order linear unconditionally energy-stable scheme for the phase field crystal equation

This paper presents a linear implicit scheme for numerically solving the phase field crystal (PFC) equation. The scheme is constructed by the scalar auxiliary variable (SAV) approach and the leapfrog scheme in temporal direction. The proposed scheme is decoupled, and it is easy to be implemented. In contrast, the previous SAV-type schemes for the PFC equation are usually coupled. The scheme satisfies the energy stability at the discrete level. Moreover, it is proved that the scheme is convergent with second-order accuracy in the temporal direction. Several numerical examples are carried out to confirm the theoretical results.

A Linearized Implicit Scheme for Solving Logarithmic Nonlinear Schrödinger’s Equation

In this research, we will develop a numerical scheme for solving the nonlinear logarithmic Schrödinger’s equation using the Linearized implicit scheme, that has second-order accuracy in space and time, with significant savings in computational time. We then compare the results to those obtained previously using the Crank-Nicolson scheme of the finite difference method. The stability and precision of this method will be evaluated before it is implemented. Precise solution and conserved quantities will be used to prove the efficiency as well as reliability of the method that has been proposed. Additionally, tests are conducted to explore the interactions that take place between two and three solitons. The numerical findings of our investigation demonstrated that the behavior of interactions is elastic.

Implicit discretization of Lagrangian gas dynamics

We construct an original framework based on convex analysis to prove the existence and uniqueness of a solution to a class of implicit numerical schemes. We propose an application of this general framework in the case of a new non linear implicit scheme for the 1D Lagrangian gas dynamics equations. We provide numerical illustrations that corroborate our proof of unconditional stability for this non linear implicit scheme.

Balanced implicit methods with strong order 1.5 for solving stochastic differential equations

The aim of this article is to propose a family of balanced implicit methods and split-step balanced implicit methods for solving Itô stochastic differential equations. These numerical methods represent a class of highly efficient linear-implicit schemes which covered the implicitness in the stochastic terms. The convergence in mean-square with strong order 1.5 and the asymptotic mean-square stability properties of these methods are analysed. What is more, the linear mean-square stability regions of some proposed methods are given. Several numerical examples are carried out to illustrate the theoretic findings.

Numerical analysis for solving Allen-Cahn equation in 1D and 2D based on higher-order compact structure-preserving difference scheme

In this paper, we present a fourth-order difference scheme for solving the Allen-Cahn equation in both 1D and 2D. The proposed scheme is described by the compact difference operators together with the additional stabilized term. As a matter of fact, the Allen-Cahn equation contains the nonlinear reaction term which is eminently proved that numerical schemes are mostly nonlinear. To solve the complexity of nonlinearity, the Crank-Nicolson/Adams-Bashforth method is applied in order to deal with the nonlinear terms with the linear implicit scheme. The well-known energy-decaying property of the equation is maintained by the proposed scheme in the discrete sense. Additionally, the L∞ error analysis is carried out in the 1D case in a rigorous way to show that the method is fourth-order and second-order accuracy for the spatial and temporal step sizes, respectively. Concurrently, we examine the L2 and H1 error analysis for the scheme in the case of 2D. We consider the impact of the additional stabilized term on numerical solutions. The consequences confirm that an appropriate value of the stabilized term yields a significant improvement. Moreover, relevant results are carried out in the numerical simulations to illustrate the faithfulness of the present method by the confirmation of existing pieces of evidence.

Scalar Auxiliary Variable/Lagrange multiplier based pseudospectral schemes for the dynamics of nonlinear Schrödinger/Gross-Pitaevskii equations

In this paper, based on the Scalar Auxiliary Variable (SAV) approach [44,45] and a newly proposed Lagrange multiplier (LagM) approach [22,21] originally constructed for gradient flows, we propose two linear implicit pseudo-spectral schemes for simulating the dynamics of general nonlinear Schrödinger/Gross-Pitaevskii equations. Both schemes are of spectral/second-order accuracy in spatial/temporal direction. The SAV based scheme preserves a modified total energy and approximate the mass to third order (with respect to time steps), while the LagM based scheme could preserve exactly the mass and original total energy. A nonlinear algebraic system has to be solved at every time step for the LagM based scheme, hence the SAV scheme is usually more efficient than the LagM one. On the other hand, the LagM scheme may outperform the SAV ones in the sense that it conserves the original total energy and mass and usually admits smaller errors. Ample numerical results are presented to show the effectiveness, accuracy and performance of the proposed schemes.

Linearized Implicit Methods Based on a Single-Layer Neural Network: Application to Keller–Segel Models

This paper is concerned with numerical approximation of some two-dimensional Keller–Segel chemotaxis models, especially those generating pattern formations. The numerical resolution of such nonlinear parabolic–parabolic or parabolic–elliptic systems of partial differential equations consumes a significant computational time when solved with fully implicit schemes. Standard linearized semi-implicit schemes, however, require reasonable computational time, but suffer from lack of accuracy. In this work, two methods based on a single-layer neural network are developed to build linearized implicit schemes: a basic one called the each step training linearized implicit method and a more efficient one, the selected steps training linearized implicit method. The proposed schemes, which make use also of a spatial finite volume method with a hybrid difference scheme approximation for convection–diffusion fluxes, are first derived for a chemotaxis system arising in embryology. The convergence of the numerical solutions to a corresponding weak solution of the studied system is established. Then the proposed methods are applied to a number of chemotaxis models, and several numerical tests are performed to illustrate their accuracy, efficiency and robustness. Generalization of the developed methods to other nonlinear partial differential equations is straightforward.

Convergence analysis of the higher-order global mass-preserving numerical method for the symmetric regularized long-wave equation

A higher-order uncoupled finite difference scheme is proposed and analysed to approximate the solutions of the symmetric regularized long-wave equation. The finite difference technique preserving the global conservation laws precisely on any time-space regions gives a three-level linear-implicit scheme with a tridiagonal system. The existence and uniqueness of numerical solutions are guaranteed while the convergence and stability are verified. In addition, the error estimation in -norm for the proposed scheme is examined, and the spatial accuracy is analysed and found to be fourth order on a uniform grid. The scheme is also proved to conserve mass and bound of solutions. Some numerical tests are presented to illustrate the theoretical results and the efficiency of the scheme. The consequences confirm that the proposed scheme gives an improvement over existing schemes. Moreover, in the numerical simulations, the faithfulness of the proposed method is validated by the evidences of an overtaking collision between two elevation solitary waves and a head-on collision between elevation as well as depression solitary waves under the effect of variable parameters.

A Preconditioning Technique for All-at-Once System from the Nonlinear Tempered Fractional Diffusion Equation

An all-at-once linear system arising from the nonlinear tempered fractional diffusion equation with variable coefficients is studied. Firstly, the nonlinear and linearized implicit schemes are proposed to approximate such the nonlinear equation with continuous/discontinuous coefficients. The stabilities and convergences of the two schemes are proved under several suitable assumptions, and numerical examples show that the convergence orders of these two schemes are $1$ in both time and space. Secondly, a nonlinear all-at-once system is derived based on the nonlinear implicit scheme, which may suitable for parallel computations. Newton's method, whose initial value is obtained by interpolating the solution of the linearized implicit scheme on the coarse space, is chosen to solve such the nonlinear all-at-once system. To accelerate the speed of solving the Jacobian equations appeared in Newton's method, a robust preconditioner is developed and analyzed. Numerical examples are reported to demonstrate the effectiveness of our proposed preconditioner. Meanwhile, they also imply that such the initial guess for Newton's method is more suitable.

Numerical Simulation of Modified Kortweg-de Vries Equation by Linearized Implicit Schemes

In this paper, we are going to derive four numerical methods for solving the Modified Kortweg-de Vries (MKdV) equation using fourth Pade approximation for space direction and Crank Nicolson in the time direction. Two nonlinear schemes and two linearized schemes are presented. All resulting schemes will be analyzed for accuracy and stability. The exact solution and the conserved quantities are used to highlight the efficiency and the robustness of the proposed schemes. Interaction of two and three solitons will be also conducted. The numerical results show that the interaction behavior is elastic and the conserved quantities are conserved exactly, and this is a good indication of the reliability of the schemes which we derived. A comparison with some existing is presented as well.

Linear-implicit and energy-preserving schemes for the Benjamin-type equations

The Benjamin-type equations are typical types of non-local partial differential equations usually describing long internal waves along the interface of two vigorously different fluid layers. In this work, we propose two kinds of novel linear-implicit and energy-preserving algorithms for the Benjamin-type equations. These algorithms are based on the invariant energy quadratization (IEQ) and scalar auxiliary variable (SAV) approaches, respectively. The IEQ and SAV are originally developed to construct energy stable schemes for the class of gradient flows. Herein, we innovate such schemes to the Benjamin-type equations and, essentially, verify them to be effective to construct energy-preserving schemes for the Hamiltonian structures. Meanwhile, numerical experiments are presented to demonstrate the efficiency of these schemes eventually.

Two novel linear-implicit momentum-conserving schemes for the fractional Korteweg-de Vries equation

We propose two conservative linear-implicit schemes for the space fractional Korteweg-de Vries (fKdV) equation. One is the linear-implicit Crank–Nicolson scheme and the other is the linear-implicit leap-frog scheme. In order to obtain a high order discretization in the space direction, we adopt the Fourier pseudospectral method. The Crank–Nicolson scheme and leap-frog scheme are used for temporal discretization, and those two schemes are efficient in practical computations because of their linear property. Furthermore, we analyse the uniqueness, boundness, convergence of the two schemes. Numerical experiments are presented to validate the theoretical analysis.

An efficient energy-preserving algorithm for the Lorentz force system

In this paper, by utilizing the invariant energy quadratization method to transform the original Hamiltonian energy functional into a quadratic form, we propose a novel energy-preserving scheme to solve the Lorentz force system. The method is a linear-implicit scheme for canonical Hamiltonian system, and can efficiently simulate the motion of charged particles in constant magnetic field. With comparison to the well-used Boris method and a similar energy-preserving BDLI method [26], numerical experiments are presented to demonstrate the energy-preserving property and computational efficiency of the method.

Study of an asymptotic preserving scheme for the quasi neutral Euler–Boltzmann model in the drift regime

We deal with the numerical approximation of a simplified quasi neutral plasma model in the drift regime. Specifically, we analyze a finite volume scheme for the quasi neutral Euler–Boltzmann equations. We prove the unconditional stability of the scheme and give some bounds on the numerical approximation that are uniform in the asymptotic parameter. The proof relies on the control of the positivity and the decay of a discrete energy. The severe non linearity of the scheme being the price to pay to get the unconditional stability, to solve it, we propose an iterative linear implicit scheme that reduces to an elliptic system. The elliptic system enjoys a maximum principle that enables to prove the conservation of the positivity under a CFL condition that does not involve the asymptotic parameter. The linear L2 stability analysis of the iterative scheme shows that it does not request the mesh size and time step to be smaller than the asymptotic parameter. Numerical illustrations are given to illustrate the stability and consistency of the scheme in the drift regime as well as its ability to compute correct shock speeds.

Second Order Fully Discrete Energy Stable Methods on Staggered Grids for Hydrodynamic Phase Field Models of Binary Viscous Fluids

We present second order, fully discrete, energy stable methods on spatially staggered grids for a hydrodynamic phase field model of binary viscous fluid mixtures in a confined geometry subject to both physical and periodic boundary conditions. We apply the energy quadratization strategy to develop a linear-implicit scheme. We then extend it to a decoupled, linear scheme by introducing an intermediate velocity term so that the phase variable, velocity field, and pressure can be solved sequentially. The two new, fully discrete linear schemes are then shown to be unconditionally energy stable, and the linear systems resulting from the schemes are proved uniquely solvable. Rates of convergence of the two linear schemes in both space and time are verified numerically. The decoupled scheme tends to introduce excessive dissipation compared to the coupled one. The coupled scheme is then used to simulate fluid drops of one fluid in the matrix of another fluid as well as mixing dynamics of binary polymeric, viscous solutions. The numerical results in mixing dynamics reveals the dramatic difference between the morphology in the simulations obtained using the two different boundary conditions (physical vs. periodic), demonstrating the importance of using proper boundary conditions in fluid dynamics simulations.

Energy Stability Analysis of Some Fully Discrete Numerical Schemes for Incompressible Navier–Stokes Equations on Staggered Grids

In this paper we consider the energy stability estimates for some fully discrete schemes which both consider time and spatial discretizations for the incompressible Navier–Stokes equations. We focus on three kinds of fully discrete schemes, i.e., the linear implicit scheme for time discretization with the finite difference method (FDM) on staggered grids for spatial discretization, pressure-correction schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations, and pressure-stabilization schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations. The energy stability estimates are obtained for the above each fully discrete scheme. The upwind scheme is used in the discretization of the convection term which plays an important role in the design of unconditionally stable discrete schemes. Numerical results are given to verify the theoretical analysis.

ASIS v1.0: an adaptive solver for the simulation of atmospheric chemistry

Abstract. This article reports on the development and tests of the adaptive semi-implicit scheme (ASIS) solver for the simulation of atmospheric chemistry. To solve the ordinary differential equation systems associated with the time evolution of the species concentrations, ASIS adopts a one-step linearized implicit scheme with specific treatments of the Jacobian of the chemical fluxes. It conserves mass and has a time-stepping module to control the accuracy of the numerical solution. In idealized box-model simulations, ASIS gives results similar to the higher-order implicit schemes derived from the Rosenbrock's and Gear's methods and requires less computation and run time at the moderate precision required for atmospheric applications. When implemented in the MOCAGE chemical transport model and the Laboratoire de Météorologie Dynamique Mars general circulation model, the ASIS solver performs well and reveals weaknesses and limitations of the original semi-implicit solvers used by these two models. ASIS can be easily adapted to various chemical schemes and further developments are foreseen to increase its computational efficiency, and to include the computation of the concentrations of the species in aqueous-phase in addition to gas-phase chemistry.

Robust and real-time guidewire simulation based on Kirchhoff elastic rod for vascular intervention training

Virtual reality (VR) based vascular intervention training is a fascinating innovation, which helps trainees develop skills in safety remote from patients. The vascular intervention training involves the use of flexible tipped guidewires to advance diagnostic or therapeutic catheters into a patient’s vascular anatomy. In this paper, a real-time physically-based modeling approach is proposed to simulate complicated behaviors of guidewires and catheters based on Kirchhoff elastic rod. The slender body of guidewire and catheter is simulated using more efficient special case of naturally straight, isotropic Kirchhoff rods, and the short flexible tip composed of straight or angled design is modeled using more complex generalized Kirchhoff rods. We derive the equations of motion for guidewire and catheter with continuous elastic energy, and then they were discretized using a linear implicit scheme that guarantees stability and robustness. In addition, we apply a fast-projection method to enforce the inextensibility of guidewire and catheter, while an adaptive sampling algorithm is implemented to improve the simulation efficiency without reducing accuracy. Experimental results reveal that our guidewire simulation method is both robust and efficient in a real-time performance.

A robust and real-time vascular intervention simulation based on Kirchhoff elastic rod

A virtual reality (VR) based vascular intervention simulation system is introduced in this paper, which helps trainees develop surgical skills and experience complications in safety remote from patients. The system simulates interventional radiology procedures, in which flexible tipped guidewires are employed to advance diagnostic or therapeutic catheters into vascular anatomy of a patient. A real-time physically-based modeling approach ground on Kirchhoff elastic rod is proposed to simulate complicated behaviors of guidewires and catheters. The slender body of guidewire and catheter is modeled using more efficient special case of naturally straight, isotropic Kirchhoff rods, and the shorter flexible tip composed of straight or angled design is modeled using more complex generalized Kirchhoff rods. The motion equations for guidewire and catheter were derived with continuous elastic energy, followed by a discretization using a linear implicit scheme that guarantees stability and robustness. In addition, we used a fast-projection method to enforce the inextensibility of guidewire and catheter. An adaptive sampling algorithm was also implemented to improve the simulation efficiency without decrease of accuracy. Experimental results revealed that our system is both robust and efficient in a real-time performance.