Fourth-order Accuracy In Time Research Articles

Algebraically stable SDIRK methods with controllable numerical dissipation for first/second-order time-dependent problems

In this paper, a family of four-stage singly diagonally implicit Runge-Kutta methods are proposed to solve first-/second-order time-dependent problems, exhibiting the following numerical properties: fourth-order accuracy in time, unconditional stability, controllable numerical dissipation, and adaptive time step selection. The BN-stability condition is employed as a constraint to optimize parameters in the Butcher table, having significant benefits, and hence is recommended for nonlinear dynamics problems in contrast to existing methods. Numerical examples involving both first- and second-order linear/nonlinear dynamics problems validate the proposed method, and numerical results reveal that the proposed methods are free from the order reduction phenomenon when applied to nonlinear dynamics problems. The performance of adaptive time-stepping using the embedded scheme is further illustrated by the phase-field modeling problem. Additionally, the advantages and disadvantages of three-stage third-order accurate algebraically stable methods are discussed. The proposed high-order time integration can be readily integrated into high-order spatial discretization methods, such as the high-order spectral element method employed in this paper, to obtain high-order discretization in space and time dimensions.

Error analysis of the exponential wave integrator sine pseudo-spectral method for the higher-order Boussinesq equation

AbstractIn this paper, we derive a new exponential wave integrator sine pseudo-spectral (EWI-SP) method for the higher-order Boussinesq equation involving the higher-order effects of dispersion. The method is fully-explicit and it has fourth order accuracy in time and spectral accuracy in space. We rigorously carry out error analysis and establish error bounds in the Sobolev spaces. The performance of the EWI-SP method is illustrated by examining the long-time evolution of the single solitary wave, single wave splitting, and head-on collision of solitary waves. Numerical experiments confirm the theoretical results.

On the Kadomtsev–Petviashvili equation with double-power nonlinearities

In this paper, we study the generalized KP equation with double-power nonlinearities. Our investigation covers various aspects, including the existence of solitary waves, their nonlinear stability, and instability. Notably, we address a broader class of nonlinearities represented by μ1|u|p1−1u+μ2|u|p2−1u, with p1>p2, encompassing cases where μ1>0 and μ1<0<μ2. One of the distinct features of our work is the absence of scaling, which introduces several challenges in establishing the existence of ground states. To overcome these challenges, we employ two different minimization problems, offering novel approaches to address this issue. Furthermore, our study includes a nuanced analysis to ascertain the stability of these ground states. Intriguingly, we extend our stability analysis to encompass cases where the convexity of the Lyapunov function is not guaranteed. This expansion of stability criteria represents a significant contribution to the field. Moving beyond the analysis of solitary waves, we shift our focus to the associated Cauchy problem. Here, we derive criteria that determine whether solutions exhibit finite-time blow-up or remain uniformly bounded within the energy space. Remarkably, our study unveils a notable gap in the existing literature, characterized by the absence of both theoretical evidence of blow-up and uniform boundedness. To explore this intriguing scenario, we employ the integrating factor method, providing a numerical investigation of solution behavior. This method distinguishes itself by offering spectral-order accuracy in space and fourth-order accuracy in time. Lastly, we rigorously establish the strong instability of the ground states, adding another layer of understanding to the complex dynamics inherent in the generalized KP equation.

Sixth-order finite difference schemes for nonlinear wave equations with variable coefficients in three dimensions

First, a nonlinear difference scheme is proposed to solve the three-dimensional (3D) nonlinear wave equation by combining the correction technique of truncation error remainder in time and a sixth-order finite difference operator in space, resulting in fourth-order accuracy in time and sixth-order accuracy in space. Then, the Richardson extrapolation method is applied to improve the temporal accuracy from the fourth-order to the sixth-order. To enhance computational efficiency, a linearized difference scheme is obtained by linear interpolation based on the nonlinear scheme. In addition, the stability of the linearized scheme is proved. Finally, the accuracy, stability and efficiency of the two proposed schemes are tested numerically.

Algorithms for coupled Burgers’ equations by sharing characteristic curves within BSLM

This paper introduces a new perspective of the traditional view on the velocity of each physical particle in the coupled Burgers’ equation in the backward semi-Lagrangian method (BSLM). The proposed methods reduce the number of Cauchy problems to be solved by observing a single virtual characteristic curve with a velocity. This can drastically reduce the computational cost of determining the departure point. Then, we solve the derived system reflected by the single virtual characteristic curve. Moreover, an efficient strategy for the derived linear system of equations is provided. Four examples are tested to demonstrate the adaptability and efficiency of the proposed method. The test results show that the proposed method has third- and fourth-order accuracy in time and space, respectively. In addition, compared with the existing method of solving the problem along two particles with different velocities, we confirm that the proposed method significantly reduces computational cost while maintaining accuracy well.

A compact ADI finite difference method for 2D reaction–diffusion equations with variable diffusion coefficients

Reaction–diffusion systems on a spatially heterogeneous domain have been widely used to model various biological applications. However, solving such partial differential equations (PDEs) analytically is rarely possible. Therefore, efficient and accurate numerical methods for solving such PDEs are desired. In this paper, we apply the well-known Padé approximation-based operator splitting techniques and develop a fourth-order compact alternative directional implicit (ADI) scheme. The new scheme is compact and fourth-order accurate in space. Combined with the Richardson extrapolation, the method can be improved to fourth-order accuracy in time. Stability analysis shows that the method is unconditionally stable; thus, a large time step can be used to improve the overall computational efficiency. Numerical examples have also demonstrated the new scheme’s high efficiency and high order accuracy.

Energy analysis of the ADI-FDTD method with fourth-order accuracy in time for Maxwell's equations

&lt;abstract&gt;&lt;p&gt;In this work, the ADI-FDTD method with fourth-order accuracy in time for the 2-D Maxwell's equations without sources and charges is proposed. We mainly focus on energy analysis of the proposed ADI-FDTD method. By using the energy method, we derive the numerical energy identity of the ADI-FDTD method and show that the ADI-FDTD method is approximately energy-preserving. In comparison with the energy in theory, the numerical one has two perturbation terms and can be used in computation in order to keep it approximately energy-preserving. Numerical experiments are given to show the performance of the proposed ADI-FDTD method which confirm the theoretical results.&lt;/p&gt;&lt;/abstract&gt;

Fast Compact Difference Scheme for Solving the Two-Dimensional Time-Fractional Cattaneo Equation

The time-fractional Cattaneo equation is an equation where the fractional order α∈(1,2) has the capacity to model the anomalous dynamics of physical diffusion processes. In this paper, we consider an efficient scheme for solving such an equation in two space dimensions. First, we obtain the space’s semi-discrete numerical scheme by using the compact difference operator in the spatial direction. Then, the semi-discrete scheme is converted to a low-order system by means of order reduction, and the fully discrete compact difference scheme is presented by applying the L2-1σ formula. To improve the computational efficiency, we adopt the fast discrete Sine transform and sum-of-exponentials techniques for the compact difference operator and L2-1σ difference operator, respectively, and derive the improved scheme with fast computations in both time and space. That aside, we also consider the graded meshes in the time direction to efficiently handle the weak singularity of the solution at the initial time. The stability and convergence of the numerical scheme under the uniform meshes are rigorously proven, and it is shown that the scheme has second-order and fourth-order accuracy in time and in space, respectively. Finally, numerical examples with high-dimensional problems are demonstrated to verify the accuracy and computational efficiency of the derived scheme.

A higher-order accurate operator splitting spectral method for the Wigner–Poisson system

An accurate description of 2-D quantum transport in a double-gate metal oxide semiconductor field effect transistor (dgMOSFET) requires a high-resolution solver for a coupled system of the 4-D Wigner equation and 2-D Poisson equation. In this paper, we propose an operator-splitting spectral method to evolve such Wigner–Poisson (WP) system in 4-D phase space with high accuracy. After the operator splitting of the Wigner equation, the resulting two sub-equations can be solved analytically with spectral approximation in phase space. Meanwhile, we adopt a Chebyshev spectral method to solve the Poisson equation. Spectral convergence in phase space and a fourth-order accuracy in time are both numerically verified. Finally, we apply the proposed solver to the simulation of a dgMOSFET, develop the steady states via long-time simulations and obtain numerically converged current–voltage (I–V) curves.

Conservative compact difference scheme for the generalized Korteweg-de Vries equation

In this paper, a linear mass and energy conservative finite difference scheme for the generalized Korteweg-de Vries (GKdV) equation is proposed and analyzed. The scheme is three-level and linear implicit and gives second- and fourth-order accuracy in time and space, respectively. It is rigorously proved by using the discrete energy method that the proposed difference scheme is uniquely solvable, stable and convergent. Numerical examples are given to confirm the stability and convergence of the numerical solution with fourth-order accuracy and the effectiveness of the present scheme for handling the single and multi-solitary waves for a long time.

Explicit high-order exponential time integrator for discontinuous Galerkin solution of Maxwell's equations

In this paper, exponential propagator (EP) is introduced to the discontinuous Galerkin finite element method (DG-FEM) for solving transient electromagnetic problems. As a kind of explicit time-stepping method, the proposed EP has fourth order accuracy in time and requires only one storage unit for each of the degrees-of-freedom. It provides an attractive alternative to the high-order low-storage explicit Runge-Kutta (LSERK) approach, which is the most popular time-marching scheme currently used in high-fidelity simulation of Maxwell's equations with DG-FEM. When centered numerical flux is utilized, the discretized Maxwell's system in lossless region is symplectic structure-preserving (energy conserving), and thus very suitable for accurately computing the long-time propagation of waves in non-dissipative medium, or simulating the motion of charged particles in electromagnetic fields. Our method can also handle upwind numerical flux at the sacrifice of symplectic-conserving virtue for higher accuracy in spatial discretization and spurious inhibition. A major contribution of this work is that, the application range of this scheme is extended to lossy media such as conductors and particularly unsplit perfectly matched layers in the framework of DG-FEM. This greatly facilitates the treatment of open boundary conditions in scattering and/or radiation problems. Extensive numerical examples, including the analysis of photonic crystals, dielectric coated perfect conducting cylinder, propagation of waves in zero-index materials, and corrugated O-type Cherenkov oscillators etc., are presented to demonstrate its stability, accuracy, and performance. All the numerical experiments show that the presented method is superior to the LSERK scheme when centered numerical flux is adopted.

Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations

&lt;p style='text-indent:20px;'&gt;The stability and convergence of the Fourier pseudo-spectral method are analyzed for the three dimensional incompressible Navier-Stokes equation, coupled with a variety of time-stepping methods, of up to fourth order temporal accuracy. An aliasing error control technique is applied in the error estimate for the nonlinear convection term, while an a-priori assumption for the numerical solution at the previous time steps will also play an important role in the analysis. In addition, a few multi-step temporal discretization is applied to achieve higher order temporal accuracy, while the numerical stability is preserved. These semi-implicit numerical schemes use a combination of explicit Adams-Bashforth extrapolation for the nonlinear convection term, as well as the pressure gradient term, and implicit Adams-Moulton interpolation for the viscous diffusion term, up to the fourth order accuracy in time. Optimal rate convergence analysis and error estimates are established in details. It is proved that, the Fourier pseudo-spectral method coupled with the carefully designed time-discretization is stable provided only that the time-step and spatial grid-size are bounded by two constants over a finite time. Some numerical results are also presented to verify the established convergence rates of the proposed schemes.&lt;/p&gt;

Exponential Runge–Kutta Method for Two-Dimensional Nonlinear Fractional Complex Ginzburg–Landau Equations

In this work, we study numerically two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equations. A centered finite difference method is exploited to discretize the spatial variables and leads to a system of the ordinary differential equation, in which the resulting coefficient matrix is complex symmetric and possesses the block Toeplitz structure. An exponential Runge–Kutta method is employed to solve such a system of the ordinary differential equation. Theoretically, the proposed method is second-order accuracy in space and fourth-order accuracy in time, respectively. In the practical implementation, the product of a block Toeplitz matrix exponential and a vector is calculated by the shift-invert Lanczos method. Meanwhile, the sectorial operator (the coefficient matrix) guarantees the fast approximation by the shift-invert Lanczos method. Numerical experiments are carried out to testify the theoretical results and demonstrate that the proposed method enjoys the excellent computational advantage.

Time Domain Room Acoustic Solver with Fourth-Order Explicit FEM Using Modified Time Integration

This paper presents a proposal of a time domain room acoustic solver using novel fourth-order accurate explicit time domain finite element method (TD-FEM), with demonstration of its applicability for practical room acoustic problems. Although time domain wave acoustic methods have been extremely attractive in recent years as room acoustic design tools, a computationally efficient solver is demanded to reduce their overly large computational costs for practical applications. Earlier, the authors proposed an efficient room acoustic solver using explicit TD-FEM having fourth-order accuracy in both space and time using low-order discretization techniques. Nevertheless, this conventional method only achieves fourth-order accuracy in time when using only square or cubic elements. That achievement markedly impairs the benefits of FEM with geometrical flexibility. As described herein, that difficulty is solved by construction of a specially designed time-integration method for time discretization. The proposed method can use irregularly shaped elements while maintaining fourth-order accuracy in time without additional computational complexity compared to the conventional method. The dispersion and dissipation characteristics of the proposed method are examined respectively both theoretically and numerically. Moreover, the practicality of the method for solving room acoustic problems at kilohertz frequencies is presented via two numerical examples of acoustic simulations in a rectangular sound field including complex sound diffusers and in a complexly shaped concert hall.

A fourth-order adaptive mesh refinement algorithm for the multicomponent, reacting compressible Navier–Stokes equations

In this paper, we present a fourth-order in space and time block-structured adaptive mesh refinement algorithm for the compressible multicomponent reacting Navier–Stokes equations. The algorithm uses a finite-volume approach that incorporates a fourth-order discretisation of the convective terms. The time-stepping algorithm is based on a multi-level spectral deferred corrections method that enables explicit treatment of advection and diffusion coupled with an implicit treatment of reactions. The temporal scheme is embedded in a block-structured adaptive mesh refinement algorithm that includes subcycling in time with spectral deferred correction sweeps applied on levels. Here we present the details of the multi-level scheme paying particular attention to the treatment of coarse–fine boundaries required to maintain fourth-order accuracy in time. We then demonstrate the convergence properties of the algorithm on several test cases including both non-reacting and reacting flows. Finally we present simulations of a vitiated dimethyl ether jet in 2D and a turbulent hydrogen jet in 3D, both with detailed kinetics and transport.

A Fully Fourth Order Accurate Energy Stable Finite Difference Method for Maxwell's Equations in Metamaterials

We present a novel fully fourth order in time and space finite difference method (FDM) for the time domain Maxwell's equations in metamaterials. We consider a Drude metamaterial model for the material response to incident electromagnetic fields. We consider the second-order formulation of the system of partial differential equations that govern the evolution in time of electric and magnetic fields along with the evolution of the polarization and magnetization current densities. Our discretization employs fourth-order staggering in space of different field components and the modified equation approach to obtain fourth-order accuracy in time. Using the energy method, we derive energy relations for the continuous models and design numerical schemes that preserve a discrete analogue of the energy relation. Numerical simulations are provided in one- and two-dimensional settings to illustrate fourth-order convergence as well as to compare with second-order schemes.

A High Order Finite Difference Method for Tempered Fractional Diffusion Equations with Applications to the CGMY Model

Using the weighted and shifted Lubich difference (WSLD) operator, we propose an efficient and stable difference scheme for solving space tempered fractional diffusion equations. The scheme has fourth order accuracy in space and second order accuracy in time. To improve the computational efficiency further, a fourth order backward Euler scheme is used to achieve fourth order accuracy in time. The fourth order backward Euler method is shown to be stable under a minor requirement. For two-dimensional problems, the Crank--Nicolson alternating direction implicit (CN-ADI) scheme is used to reduce computational complexity. These proposed methods are applied to the Carr--Geman--Madan--Yor (CGMY) model in finance, and several numerical experiments are designed to verify the efficiency of the proposed methods.

Fourth-Order Time-Stepping For Stiff PDEs On The Sphere

We present in this paper algorithms for solving stiff PDEs on the unit sphere with spectral accuracy in space and fourth-order accuracy in time. These are based on a variant of the double Fourier sphere method in coefficient space with multiplication matrices that differ from the usual ones, and implicit-explicit time-stepping schemes. Operating in coefficient space with these new matrices allows one to use a sparse direct solver, avoids the coordinate singularity and maintains smoothness at the poles, while implicit-explicit schemes circumvent severe restrictions on the time-steps due to stiffness. A comparison is made against exponential integrators and it is found that implicit-explicit schemes perform best. Implementations in MATLAB and Chebfun make it possible to compute the solution of many PDEs to high accuracy in a very convenient fashion.

Efficient and accurate numerical simulation of acoustic wave propagation in a 2D heterogeneous media

In this paper, a compact fourth-order finite difference scheme is derived to solve the 2D acoustic wave equation in heterogenous media. The Padé approximation is used to obtain fourth-order accuracy in both temporal and spatial dimensions, and the alternating direction implicit (ADI) technique is used to reduce the computational cost. Due to the non-constant wave velocity, the conventional ADI method is hard to implement as the algebraic manipulation cannot be used here. A novel numerical strategy is proposed in this work so that the compact scheme still maintains fourth-order accuracy in time and space. The fourth-order convergence order was firstly proved by theoretical error analysis, then was confirmed by numerical examples. It was shown that the proposed method is conditionally stable with a Courant–Friedrichs–Lewy (CFL) condition that is comparable to other existing finite difference schemes. Several numerical examples were solved to demonstrate the efficiency and accuracy of the new algorithm.

The Piecewise Cubic Method (PCM) for computational fluid dynamics

We present a new high-order finite volume reconstruction method for hyperbolic conservation laws. The method is based on a piecewise cubic polynomial which provides its solutions a fifth-order accuracy in space. The spatially reconstructed solutions are evolved in time with a fourth-order accuracy by tracing the characteristics of the cubic polynomials. As a result, our temporal update scheme provides a significantly simpler and computationally more efficient approach in achieving fourth order accuracy in time, relative to the comparable fourth-order Runge–Kutta method. We demonstrate that the solutions of PCM converges at fifth-order in solving 1D smooth flows described by hyperbolic conservation laws. We test the new scheme on a range of numerical experiments, including both gas dynamics and magnetohydrodynamics applications in multiple spatial dimensions.