Crank-Nicolson Leap-Frog Research Articles

Spectral Galerkin Methods for Riesz Space-Fractional Convection–Diffusion Equations

This paper applies the spectral Galerkin method to numerically solve Riesz space-fractional convection–diffusion equations. Firstly, spectral Galerkin algorithms were developed for one-dimensional Riesz space-fractional convection–diffusion equations. The equations were solved by discretizing in space using the Galerkin–Legendre spectral approaches and in time using the Crank–Nicolson Leap-Frog (CNLF) scheme. In addition, the stability and convergence of semi-discrete and fully discrete schemes were analyzed. Secondly, we established a fully discrete form for the two-dimensional case with an additional complementary term on the left and then obtained the stability and convergence results for it. Finally, numerical simulations were performed, and the results demonstrate the effectiveness of our numerical methods.

A second-order in time, BGN-based parametric finite element method for geometric flows of curves

Over the last two decades, the field of geometric curve evolutions has attracted significant attention from scientific computing. One of the most popular numerical methods for solving geometric flows is the so-called BGN scheme, which was proposed by Barrett et al. (2007) [8], due to its favorable properties (e.g., its computational efficiency and the good mesh property). However, the BGN scheme is limited to first-order accuracy in time, and how to develop a higher-order numerical scheme is challenging. In this paper, we propose a fully discrete, temporal second-order parametric finite element method, which integrates with two different mesh regularization techniques, for solving geometric flows of curves. The scheme is constructed based on the BGN formulation and a semi-implicit Crank-Nicolson leap-frog time stepping discretization as well as a linear finite element approximation in space. More importantly, we point out that the shape metrics, such as manifold distance and Hausdorff distance, instead of function norms, should be employed to measure numerical errors. Extensive numerical experiments demonstrate that the proposed BGN-based scheme is second-order accurate in time in terms of shape metrics. Moreover, by employing the classical BGN scheme as mesh regularization techniques, our proposed second-order schemes exhibit good properties with respect to the mesh distribution. In addition, an unconditional interlaced energy stability property is obtained for one of the mesh regularization techniques.

A Crank–Nicolson leap-frog scheme for the unsteady incompressible magnetohydrodynamics equations

This paper presents a Crank–Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. The spatial discretization adopts the Galerkin finite element method (FEM), and the temporal discretization employs the CNLF method for linear terms and the semi-implicit method for nonlinear terms. The first step uses Stokes style’s scheme, the second step employs the Crank–Nicolson extrapolation scheme, and others apply the CNLF scheme. We establish that the fully discrete scheme is stable and convergent when the time step is less than or equal to a positive constant. Firstly, we show the stability of the scheme by means of the mathematical induction method. Next, we focus on analyzing error estimates of the CNLF method, where the convergence order of the velocity and magnetic field reach second-order accuracy, and the pressure is the first-order convergence accuracy. Finally, the numerical examples demonstrate the optimal error estimates of the proposed algorithm.

Fully decoupled unconditionally stable Crank–Nicolson leapfrog numerical methods for the Cahn–Hilliard–Darcy system

AbstractWe develop two totally decoupled, linear and second‐order accurate numerical methods that are unconditionally energy stable for solving the Cahn–Hilliard–Darcy equations for two phase flows in porous media or in a Hele‐Shaw cell. The implicit‐explicit Crank–Nicolson leapfrog method is employed for the discretization of the Cahn–Hiliard equation to obtain linear schemes. Furthermore the artificial compression technique and pressure correction methods are utilized, respectively, so that the Cahn–Hiliard equation and the update of the Darcy pressure can be solved independently. We establish unconditionally long time stability of the schemes. Ample numerical experiments are performed to demonstrate the accuracy and robustness of the numerical methods, including simulations of the Rayleigh–Taylor instability, the Saffman–Taylor instability (fingering phenomenon).

An efficient energy-stable pseudospectral method for simulating vortex dynamics of the Ginzburg–Landau–Schrödinger equation

In this article, we concentrate on designing an efficient energy-stable pseudospectral method for the vortex dynamics of the Ginzburg–Landau–Schrödinger equation (GLSE) on bounded domain under homogeneous Neumann boundary condition. Based on a Lagrange multiplier (LagM) approach, we firstly introduce an equivalent system, named LagM–GLSE. Secondly, we discretize the LagM–GLSE by Crank–Nicolson Leap-Frog scheme in temporal direction. While in spatial direction, we utilize a cosine pseudospectral approach. Thus, the resulting scheme has a spectral/second-order convergence rate in space/time. The discretized scheme is efficient and easy to implement. Moreover, we prove that the LagM based scheme can dissipate/conserve the original total energy. Finally, ample numerical results are provided to show the efficiency and accuracy of our scheme. We also apply the lagM based scheme to simulate an interesting vortex dynamics with different parameters for the GLSE.

An Explicit–Implicit Spectral Element Scheme for the Nonlinear Space Fractional Schrödinger Equation

In this paper, we solve the space fractional nonlinear Schrödinger equation (SFNSE) by developing an explicit–implicit spectral element scheme, which is formulated based on the Legendre spectral element approximation in space and the Crank–Nicolson leap frog (CNLF) difference discretization in time. Both mass and energy conservative properties are discussed for the spectral element scheme. Numerical stability and convergence of the scheme are proved. Numerical experiments are performed to confirm the high accuracy and efficiency of the proposed numerical scheme.

A second order ensemble algorithm for computing the Navier-Stokes equations

We present an efficient, second order ensemble algorithm for computing the Navier-Stokes equations multiple times with different model parameters. The algorithm is based on the Crank-Nicolson Leap-frog (CNLF) scheme and the ensemble timestepping incorporating the artificial compressibility method and the recent scalar auxiliary variable (SAV) idea for developing unconditionally stable schemes for nonlinear flows. The computation of the velocity and pressure is decoupled in the algorithm resulting in smaller linear systems to be solved at each time step. All realizations of the flow corresponding to different model parameters share the same coefficient matrix so that efficient block solvers can be used to reduce the computational cost. The proposed algorithm is efficient in terms of both computer storage and CPU time. We prove the algorithm is long time stable without any timestep conditions. Ample numerical experiments are performed for various flow problems to validate the efficiency and effectiveness of the algorithm.

Unconditionally stable, second order, decoupled ensemble schemes for computing evolutionary Boussinesq equations

In this report we present two unconditionally stable, second order, decoupled ensemble schemes for computing evolutionary Boussinesq equations: the stabilized scalar auxiliary variable Crank-Nicolson leap-frog ensemble scheme (Stab-SAV-CNLF-En) and the stabilized scalar auxiliary variable BDF2 ensemble scheme (Stab-SAV-BDF2-En). The two ensemble schemes adopt the recently developed stabilized scalar auxiliary variable (SAV) idea and ensemble timestepping to achieve both high efficiency and unconditional stability. Specifically, the stabilized SAV approach makes it possible to devise unconditionally long time stable schemes for which the nonlinear terms and coupling terms in the Boussinesq equations are made fully explicit, leading to linear systems with constant coefficient matrices to be solved after spatial discretization. The ensemble timestepping further improves the efficiency by making the coefficient matrices of all realizations the same, so that efficient block solvers can be applied to solve the corresponding one linear system with multiple right hand sides and thus greatly reduce the computational cost. We prove the proposed schemes are unconditionally stable and present implementation details. Ample numerical tests are performed to show the efficiency and effectiveness of the combined approach.

Decoupled Crank–Nicolson LeapFrog method for the evolution Boussinesq equations

This paper presents decoupled second‐order accurate algorithm based on Crank–Nicolson LeapFrog (CNLF) scheme for the evolution Boussinesq equations. The proposed algorithm deals with the spatial discretization by finite element method and treat the temporal discretization by CNLF method. For the nonlinear term in the Boussinesq equations, the semi‐implicit method is used. Unconditional stability and error estimate of the numerical algorithm are proven. Some numerical tests are presented to justify the theoretical analysis.

A Fully-Decoupled Artificial Compressible Crank–Nicolson–Leapfrog Time Stepping Scheme for the Phase Field Model of Two-Phase Incompressible Flows

A Fully-Decoupled Artificial Compressible Crank–Nicolson–Leapfrog Time Stepping Scheme for the Phase Field Model of Two-Phase Incompressible Flows

A Second-Order Crank-Nicolson Leap-Frog Scheme for the Modified Phase Field Crystal Model with Long-Range Interaction.

In this paper, we construct a fully discrete and decoupled Crank-Nicolson Leap-Frog (CNLF) scheme for solving the modified phase field crystal model (MPFC) with long-range interaction. The idea of CNLF is to treat stiff terms implicity with Crank-Nicolson and to treat non-stiff terms explicitly with Leap-Frog. In addition, the scalar auxiliary variable (SAV) method is used to allow explicit treatment of the nonlinear potential, then, these technique combines with CNLF can lead to the highly efficient, fully decoupled and linear numerical scheme with constant coefficients at each time step. Furthermore, the Fourier spectral method is used for the spatial discretization. Finally, we show that the CNLF scheme is fully discrete, second-order decoupled and unconditionally stable. Ample numerical experiments in 2D and 3D are provided to demonstrate the accuracy, efficiency, and stability of the proposed method.

Three linear, unconditionally stable, second order decoupling methods for the Allen–Cahn–Navier–Stokes phase field model

Hydrodynamics coupled phase field models have intricate difficulties to solve numerically as they feature high nonlinearity and great complexity in coupling. In this paper, we propose three second order, linear, unconditionally stable decoupling methods based on the Crank–Nicolson leap-frog time discretization for solving the Allen–Cahn–Navier–Stokes (ACNS) phase field model of two-phase incompressible flows. The ACNS system is decoupled via the artificial compression method and a splitting approach by introducing an exponential scalar auxiliary variable. We prove all three algorithms are unconditionally long time stable. Numerical examples are provided to verify the convergence rate, unconditional stability, and computational efficiency.

A Triangular Spectral Element Method for the 2-D Viscous Burgers Equation

A triangular spectral element method is established for the two-dimensional viscous Burgers equation. In the spatial direction, a new type of mapping is applied. We splice the local basis functions on each triangle into a global basis function. The second-order Crank-Nicolson/ leap-frog (CNLF) method is used for discretization in the time direction. Due to the use of a quasi-interpolation operator, the nonlinear term can be handled conveniently. We give the fully discrete scheme of the method and the implementation process of the algorithm. Numerical examples verify the effectiveness of this method.

Decoupled Stabilized Crank–Nicolson LeapFrog method for time-dependent Navier–Stokes/Darcy model

In this paper, a Stabilized Crank–Nicolson LeapFrog (SCNLF) method for the non- stationary Navier–Stokes/Darcy model is presented and analyzed. Initially, we decompose the coupling model into Navier–Stokes and Darcy equations. The spatial discretization by stabilized finite element method and the temporal discretization by CNLF method give a second-order partitioned method. Stability and error estimate of the numerical method are given. Finally, some numerical tests are presented to justify the theoretical analysis.

Linearly implicit energy-preserving Fourier pseudospectral schemes for the complex modified Korteweg-de Vries equation

We propose two linearly implicit energy-preserving schemes for the complex modified Korteweg–de Vries equation, based on the invariant energy quadratization method. First, a new variable is introduced and a new Hamiltonian system is constructed for this equation. Then the Fourier pseudospectral method is used for the space discretization and the Crank–Nicolson leap-frog schemes for the time discretization. The proposed schemes are linearly implicit, which is only needed to solve a linear system at each time step. The fully discrete schemes can be shown to conserve both mass and energy in the discrete setting. Some numerical examples are also presented to validate the effectiveness of the proposed schemes. doi:10.1017/S1446181120000218

Numerical analysis of CNLF modular Grad–Div stabilization method for time-dependent Navier–Stokes equations

This paper presents a modular Grad–Div(GD) algorithm based on Crank–Nicolson LeapFrog (CNLF) scheme for the non-stationary Navier–Stokes model, which is second-order accurate. The CNLF modular Grad–Div(CNLF-MGD) method has the obvious advantage that the computational efficiency will not crash or slow down with the large parameters. Stability and error estimate of the numerical method are given. Some numerical tests are proposed to justify the theoretical analysis.

LINEARLY IMPLICIT ENERGY-PRESERVING FOURIER PSEUDOSPECTRAL SCHEMES FOR THE COMPLEX MODIFIED KORTEWEG–DE VRIES EQUATION

AbstractWe propose two linearly implicit energy-preserving schemes for the complex modified Korteweg–de Vries equation, based on the invariant energy quadratization method. First, a new variable is introduced and a new Hamiltonian system is constructed for this equation. Then the Fourier pseudospectral method is used for the space discretization and the Crank–Nicolson leap-frog schemes for the time discretization. The proposed schemes are linearly implicit, which is only needed to solve a linear system at each time step. The fully discrete schemes can be shown to conserve both mass and energy in the discrete setting. Some numerical examples are also presented to validate the effectiveness of the proposed schemes.

Crank–Nicolson Leap-Frog Time Stepping Decoupled Scheme for the Fluid–Fluid Interaction Problems

A fully discrete Crank-Nicolson leap-frog time stepping decoupled (CNLFD) scheme is presented and studied for the fluid–fluid interaction problems. The proposed scheme deals with the spatial discretization by finite element method (FEM), treats the temporal discretization by CNLF scheme and decouples the nonlinear interface condition by using a geometric averaging of the jump. The unconditional stability and error estimate are proven. Numerical tests are performed to demonstrate the robustness of this method.

A second order, linear, unconditionally stable, Crank–Nicolson–Leapfrog scheme for phase field models of two-phase incompressible flows

In this article we propose a second order, linear, unconditionally stable, implicit–explicit scheme based on the Crank–Nicolson–Leapfrog discretization and the artificial compression method for solving phase field models of two-phase incompressible flows. We show that the scheme is unconditionally long-time stable. Numerical examples are provided to demonstrate the accuracy and long-time stability.

RBF approximation by partition of unity for valuation of options under exponential Lévy processes

The prices of European and American-style contracts on assets driven by Markov processes satisfy partial integro-differential equations (PIDEs). In particular, this holds true for assets driven by Lévy processes, which are very popular in mathematical finance. We focus below on Lévy processes whose jump part has infinite (small jumps) activity, in which case the resulting singular kernel is rather difficult to treat numerically [1,23]. We illustrate, using the CGMY example, that this challenge may be successfully met by using a combination of: a) the Asmussen-Rosinsky approximation of the small jumps by Brownian motion; b) localized RBF approximations known as the RBF partition of unity (RBF-PU), which overcome the ill-conditioning inherent in global mesh-free methods; c) Crank-Nicolson Leap-Frog (CNLF) for time discretization. d) treating the local term using an implicit step, and the nonlocal term using an explicit step (to avoid the inversion of the nonsparse matrix)Finally, to deal with free boundary problems associated to American options, we combine the implicit-explicit method with a penalty method.Efficiency and practical performance are demonstrated by several numerical experiments for pricing European and American contracts.