Leapfrog Scheme Research Articles

Convergence Analysis for the Wave Equation Discretized with Hybrid Methods in Space (HHO, HDG and WG) and the Leapfrog Scheme in Time

Convergence Analysis for the Wave Equation Discretized with Hybrid Methods in Space (HHO, HDG and WG) and the Leapfrog Scheme in Time

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.

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.

Tsunami Squares: Leapfrog scheme implementation and benchmark study on wave–shore interaction of solitary waves

Impulse waves are generated by rapid subaerial mass movements including landslides, avalanches and glacier break-offs, which pose a potential risk to public facilities and residents along the shore of natural lakes or engineered reservoirs. Therefore, the prediction and assessment of impulse waves are of considerable importance to practical engineering. Tsunami Squares, as a meshless numerical method based on a hybrid Eulerian–Lagrangian algorithm, have focused on the simulation of landslide-generated impulse waves. An updated numerical scheme referred to as Tsunami Squares Leapfrog, was developed which contains a new smooth function able to achieve space and time convergence tests as well as the Leapfrog time integration method enabling second-order accuracy. The updated scheme shows improved performance due to a lower wave decay rate per unit propagation distance compared to the original implementation of Tsunami Squares. A systematic benchmark testing of the updated scheme was conducted by simulating the run-up, reflection and overland flow of solitary waves along a slope for various initial wave amplitudes, water depths and slope angles. For run-up, the updated scheme shows good performance when the initial relative wave amplitude is smaller than 0.4. Otherwise, the model tends to underestimate the run-up height for mild slopes, while an overestimation is observed for steeper slopes. With respect to overland flow, the prediction error of the maximum flow height can be limited to ± 50% within a 90% confidence interval. However, the prediction of the front propagation velocity can only be controlled to ± 100% within a 90% confidence interval. Furthermore, a sensitivity analysis of the dynamic friction coefficient of water was performed and a suggested range from 0.01 to 0.1 was given for reference.

Stiff-cut leap-frog scheme for fractional Laplacian diffusion equations

In this study, we investigate the numerical discretization of two-dimensional fractional Laplacian diffusion equations. After discretizing the space derivative, we split the discrete fractional Laplacian into two parts. One is the τ preconditioner multiplying a constant, solved using the leap-frog scheme, while the other part and the remaining nonlinear term are treated with an explicit scheme. This results in what is termed the stiff-cut leap-frog scheme, which is more efficient than the traditional leap-frog scheme. The main advantage of the proposed scheme is that the coefficient matrices of the generated linear systems belong to τ algebra, which can be diagonalized by fast discrete sine transform with the complexity of O(NlogN), where N is the freedom of unknowns. Hence, no iterative method is required at each time integration. Moreover, we theoretically prove that the proposed scheme is unconditionally stable and converges in second order. Ultimately, numerical experiments are conducted to demonstrate the precision and effectiveness of the stiff-cut leap-frog scheme.

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.

Efficient Improved Local Time Stepping with the Leapfrog Scheme for Transient 3-D Electromagnetic Analyses

An alternate boundary local time stepping (ABLTS) method is proposed for the discontinuous Galerkin time domain method for transient electromagnetic simulations to reduce the computation complexity of the local time stepping (LTS) method. The proposed method exhibites lower storage and time complexity than the conventional LF-LTS method . The stability, accuracy, and effectiveness of the ABLTS method are verified by applying it to the simulation of a resonator cavity and multi-layer microstrip antenna. The numerical results revealed that the developed method is effective for the transient electromagnetic simulation of antennas.

On the discrete equivalence of Lagrangian, Hamiltonian and mixed finite element formulations for linear wave phenomena

It is well known that the Lagrangian and Hamiltonian descriptions of field theories are equivalent at the discrete time level when variational integrators are used. Besides the symplectic Hamiltonian structure, many physical systems exhibit a Hamiltonian structure when written in mixed form. In this contribution, the discrete equivalence of Lagrangian, symplectic Hamiltonian and mixed formulations is investigated for linear wave propagation phenomena. Under compatibility conditions between the finite elements, the Lagrangian and mixed formulations are indeed equivalent. For the time discretization the leapfrog scheme and the implicit midpoint rule are considered. In mixed methods applied to wave problems the primal variable (e.g. the displacement in mechanics or the magnetic potential in electromagnetism) is not an unknown of the problem and is reconstructed a posteriori from its time derivative. When this reconstruction is performed via the trapezoidal rule, then these time-discretization methods lead to equivalent formulations.

Semi-Implicit Computation of Fast Modes in a Scheme Integrating Slow Modes by a Leapfrog Method Based on a Selective Implicit Time Filter

Abstract A scheme for integration of atmospheric equations containing terms with differing time scales is developed. The method employs a filtered leapfrog scheme utilizing a fourth-order implicit time filter with one function evaluation per time step to compute slow-propagating phenomena such as advection and rotation. The terms involving fast-propagating modes are handled implicitly with an unconditionally stable method that permits application of larger time steps and faster computations compared to fully explicit treatment. Implementation using explicit and recurrent formulation is provided. Stability analysis demonstrates that the method is conditionally stable for any combination of frequencies involved in the slow and fast terms as they approach the origin. The implicit filter used in the method damps the computational modes without noticeably sacrificing the accuracy of the physical mode. The O[(Δt4)] accuracy for amplitude errors achieved by the implicitly filtered leapfrog is preserved in applications where terms responsible for fast propagation are integrated with a semi-implicit method. Detailed formulation of the method for soundproof nonhydrostatic anelastic equations is provided. Procedures for implementation in global spectral shallow-water models are also given. Examples comparing numerical and analytical solutions for linear gravity waves demonstrate the accuracy of the scheme. The performance is also shown in more practical nonlinear applications, where numerical solutions accomplished by the method are evaluated against those computed from a scheme where the slow terms are handled by the third-order Runge–Kutta scheme. It demonstrates that the method is able to accurately resolve fine-scale dynamics of Kelvin–Helmholtz shear instabilities, the evolution of density current, and nonlinear drifts of twin tropical cyclones.

An implicit–explicit time discretization for elastic wave propagation problems in plates

AbstractWe propose a new implicit–explicit scheme to address the challenge of modeling wave propagation within thin structures using the time‐domain finite element method. Compared to standard explicit schemes, our approach renders a time marching algorithm with a time step independent of the plate thickness and its associated discretization parameters (mesh step and order of approximation). Relying on the standard three dimensional elastodynamics equations, our strategy can be applied to any type of material, either isotropic or anisotropic, with or without discontinuities in the thickness direction. Upon the assumption of an extruded mesh of the plate‐like geometry, we show that the linear system to be solved at each time step is partially lumped thus efficiently treated. We provide numerical evidence of an adequate convergence behavior, similar to a reference solution obtained using the well‐known leapfrog scheme. Further numerical investigations show significant speed up factors compared to the same reference scheme, proving the efficiency of our approach for the configurations of interest.

An energy-conserving Fourier particle-in-cell method with asymptotic-preserving preconditioner for Vlasov-Ampère system with exact curl-free constraint

We present an efficient and accurate energy-conserving implicit particle-in-cell (PIC) algorithm for the electrostatic Vlasov system, with particular emphasis on its high robustness for simulating complex plasma systems with multiple physical scales. This method consists of several indispensable elements: (i) the reformulation of the original Vlasov-Poisson system into an equivalent Vlasov-Ampère system with divergence-/curl-free constraints; (ii) a novel structure-preserving Fourier spatial discretization, which exactly preserves these constraints at the discrete level; (iii) a preconditioned Anderson-acceleration algorithm for the solution of the highly nonlinear system; and (iv) a linearized and uniform approximation of the implicit Crank-Nicolson scheme for various Debye lengths, based on the generalized Ohm's law, which serves as an asymptotic-preserving preconditioner for the proposed method. Numerical experiments are conducted, and comparisons are made among the proposed energy-conserving scheme, the classical leapfrog scheme, and a Strang operator-splitting scheme to demonstrate the superiority of the proposed method, especially for plasma systems crossing physical scales.

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.

Estimating water levels caused by a tropical storm along the Bangladesh coast: A numerical approach

In this study, water levels resulting from the dynamic interaction of tide and surge are estimated by solving a 2-D vertically integrated shallow water equations numerically. To solve the equations on the specific 2-D grid, the explicit Leapfrog scheme is implemented, adopting a staggered Arakawa C-grid. The domain's complex land-sea interface is approximated through the stair-step method in order to employ the finite difference technique. To incorporate the complexity of the domain with a considerably high accuracy and to reduce computational cost, one-way nested grid models are embraced. The Meghna River freshwater discharge is incorporated into the innermost child model. A stable tidal regime over the region of interest is generated by applying the four vital tidal constituents, namely M2 (principal lunar semidiurnal), S2 (principal solar semidiurnal), O1 (principal lunar diurnal) and K1 (luni-solar diurnal) in the southern open boundary of the outermost model. This previously effectuated tidal regime is used as the initial state of the sea in getting total water levels due to the dynamic interaction of tide and surge. Numerical experiments are performed with the storm AILA that hit the coast of Bangladesh on May 25, 2009. The simulated results are found to closely match observed and reported data.

Time-explicit hybrid high-order method for the nonlinear acoustic wave equation

We devise a fully explicit scheme for the nonlinear acoustic wave equation in its second-order formulation in time, using the HHO method for space discretization and the leapfrog scheme for time integration. The key idea for the explicitation is an iterative procedure to approximate at each time step the static nonlinear coupling between the cell and face unknowns. This procedure is based on a splitting of the HHO stabilization operator and, in the linear case, is proved to converge for a large enough weight of the stabilization uniformly in the mesh size. Increasing the stabilization weight turns out to have a moderate impact on the CFL condition. The numerical experiments demonstrate the computational efficiency of the splitting procedure compared to a semi-implicit scheme for the static coupling between cell and face unknowns.

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.

Inertial instability and phase error in Euler forward predictor-corrector time integration schemes: Improvement of modeling Great Lakes thermal structure and circulation using FVCOM

This study investigates the inertial stability properties and phase error of numerical time integration schemes in several widely-used ocean and atmospheric models. These schemes include the most widely used centered differencing (i.e., leapfrog scheme or the 3-time step scheme at n-1, n, n+1) and 2-time step (n, n+1) 1st-order Euler forward schemes, as well as 2nd-stage and 3rd- and 4th-stage Euler predictor-corrector (PC) schemes. Previous work has proved that the leapfrog scheme is neutrally stable with respect to the Coriolis force, with perfect inertial motion preservation, an amplification factor (AF) equal to unity, and a minor overestimation of the phase speed. The 1st-order Euler forward scheme, on the other hand, is known to be unconditionally inertially unstable since its AF is always greater than unity. In this study, it is shown that 3rd- and 4th-order predictor-corrector schemes 1) are inertially stable with weak damping if the Coriolis terms are equally split to n+1 (new value) and n (old value); and 2) introduce an artificial computational mode. The inevitable phase error associated with the Coriolis parameter is analyzed in depth for all numerical schemes. Some schemes (leapfrog and 2nd-stage PC schemes) overestimate the phase speed, while the others (1st-order Euler forward, 3rd- and 4th-stage PC schemes) underestimate it. To preserve phase speed as best as possible in a numerical model, alternating a scheme that overestimates the phase speed with a scheme that underestimates the phase speed is recommended. Considering all properties investigated, the leapfrog scheme is still highly recommended for a time integration scheme. As an example, a comparison between a leapfrog scheme and a 1st-order Euler forward scheme is presented to show that the leapfrog scheme reproduces much better vertical thermal stratification and circulation in the weakly-stratified Great Lakes.

A pseudospectral implicit particle-in-cell method with exact energy and charge conservation

The standard particle-in-cell (PIC) method employs explicit finite-difference (FD) schemes, such as the leap-frog scheme, for spatial and temporal integrations. In this study, we introduce a novel approach by employing a pseudospectral method for solving the Poisson equation and adopting a fully implicit time integration technique to achieve exact energy conservation. The pseudospectral field solver offers the advantage of spectral accuracy in obtaining field solutions. Previous investigations on implicit time integration of PIC FD equations have demonstrated the capability of enforcing precise energy exchange between the field and particles, resulting in energy-conserving schemes. We prove that the exact energy conservation property can be extended to the pseudospectral scheme. Additionally, we provide a solution to ensure a pseudospectral charge continuity equation. To validate our methodology, we implement the new scheme in a 2D electrostatic PIC code and present numerical examples that confirm the theoretical findings.

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.

Unconditional optimal error estimates of a linearized mass- and energy- conservation FEM for a coupled nonlinear Schrödinger equations

In this paper, we study a linearized finite element method for a coupled nonlinear Schrödinger system (CNLS). We apply a modified leap-frog scheme for time discretization and a Galerkin finite element method for spatial discretization. We prove that this method is conservative for mass and energy in a modified form. By an error splitting technique, we split the error into two parts, the temporal error and the spatial error. First we prove the uniform boundedness in H2-norm for the solutions of the time-discrete system, and obtain the temporal error estimates. Second, we prove the pointwise uniform boundedness of the finite element solution. Then we get the optimal L2-norm error estimates without any time-step restriction. Finally, in order to confirm our theoretical analysis, we provide numerical examples to validate the convergence-order, unconditional stability and conservation for mass and energy.

An unconditionally stable artificial compression method for the time‐dependent groundwater‐surface water flows

AbstractIn this article, we propose a second order, unconditionally stable artificial compression method for the fully evolutionary Stokes/Darcy and Navier‐Stokes/Darcy equations that model the coupling surface and groundwater flows. It uncouples the surface from the groundwater flow by the Crank‐Nicolson Leapfrog scheme for the discretization in time, and through the artificial compression method without artificial pressure boundary conditions to decouple the velocity and pressure of the incompressible flow. Finally, we have verified the stability and second‐order convergence of the algorithm from theoretical analysis and numerical experiments.