High-order Accuracy In Time Research Articles

An explicit finite difference time domain method with high order accuracy in time for Maxwell's equations

An explicit finite difference time domain method with high order accuracy in time for Maxwell's equations

A rectangular staggered-grid finite-difference scheme with fourth-order temporal accuracy for pseudo-acoustic VTI modelling

ABSTRACTCurrent staggered-grid finite-difference (SFD) schemes for anisotropic wave equations can ensure only high-order spatial modelling accuracy, whereas the time discretisation accuracy is still of second-order. To improve temporal modelling accuracy, we develop a rectangular SFD scheme with fourth-order temporal accuracy to solve the first-order pseudo-acoustic wave equation in vertical transversely isotropic media. High-order temporal accuracy is achieved by applying the high-order temporal derivatives, which can be replaced with the high-order spatial derivatives to reduce the computational cost. The corresponding high-order finite-difference (FD) coefficients are generated by using Taylor-series expansion and least-squares. Dispersion and stability analyses indicate that our proposed method has higher accuracy and better stability than the conventional method. Several modelling examples confirm that our proposed scheme exhibits low temporal dispersion even for a large time step. Because of the high-order accuracy in time and space domains, our new SFD scheme allows for a larger time step and shorter operator length, which is more efficient.

An implicit spatial and high-order temporal finite difference scheme for 2D acoustic modelling

The finite difference (FD) method exhibits great superiority over other numerical methods due to its easy implementation and small computational requirement. We propose an effective FD method, characterised by implicit spatial and high-order temporal schemes, to reduce both the temporal and spatial dispersions simultaneously. For the temporal derivative, apart from the conventional second-order FD approximation, a special rhombus FD scheme is included to reach high-order accuracy in time. Compared with the Lax-Wendroff FD scheme, this scheme can achieve nearly the same temporal accuracy but requires less floating-point operation times and thus less computational cost when the same operator length is adopted. For the spatial derivatives, we adopt the implicit FD scheme to improve the spatial accuracy. Apart from the existing Taylor series expansion-based FD coefficients, we derive the least square optimisation based implicit spatial FD coefficients. Dispersion analysis and modelling examples demonstrate that, our proposed method can effectively decrease both the temporal and spatial dispersions, thus can provide more accurate wavefields.

A new high-order finite volume method for 3D elastic wave simulation on unstructured meshes

In this paper, we proposed a new efficient high-order finite volume method for 3D elastic wave simulation on unstructured tetrahedral meshes. With the relative coarse tetrahedral meshes, we make subdivision in each tetrahedron to generate a stencil for the high-order polynomial reconstruction. The subdivision algorithm guarantees the number of subelements is greater than the degrees of freedom of a complete polynomial. We perform the reconstruction on this stencil by using cell-averaged quantities based on the hierarchical orthonormal basis functions. Unlike the traditional high-order finite volume method, our new method has a very local property like DG and can be written as an inner-split computational scheme which is beneficial to reducing computational amount. Moreover, the stencil in our method is easy to generate for all tetrahedrons especially in the three-dimensional case. The resulting reconstruction matrix is invertible and remains unchanged for all tetrahedrons and thus it can be pre-computed and stored before time evolution. These special advantages facilitate the parallelization and high-order computations. We show convergence results obtained with the proposed method up to fifth order accuracy in space. The high-order accuracy in time is obtained by the Runge–Kutta method. Comparisons between numerical and analytic solutions show the proposed method can provide accurate wavefield information. Numerical simulation for a realistic model with complex topography demonstrates the effectiveness and potential applications of our method. Though the method is proposed based on the 3D elastic wave equation, it can be extended to other linear hyperbolic system.

A two-dimensional artificial viscosity technique for modelling discontinuity in shallow water flows

In this study, a two-dimensional cell-centred finite volume scheme is used to simulate discontinuity in shallow water flows. Instead of using a Riemann solver, an artificial viscosity technique is developed to minimise unphysical oscillations. This is constructed from a combination of a Laplacian and a biharmonic operator using a maximum eigenvalue of the Jacobian matrix. In order to achieve high-order accuracy in time, we use the fourth-order Runge–Kutta method. A hybrid formulation is then proposed to reduce computational time, in which the artificial viscosity technique is only performed once per time step. The convective flux of the shallow water equations is still re-evaluated four times, but only by averaging left and right states, thus making the computation much cheaper. A comparison of analytical and laboratory results shows that this method is highly accurate for dealing with discontinuous flows. As such, this artificial viscosity technique could become a promising method for solving the shallow water equations.

Implicit Multistage Two-Derivative Discontinuous Galerkin Schemes for Viscous Conservation Laws

In this paper we apply implicit two-derivative multistage time integrators to conservation laws in one and two dimensions. The one dimensional solver discretizes space with the classical discontinuous Galerkin method, and the two dimensional solver uses a hybridized discontinuous Galerkin spatial discretization for efficiency. We propose methods that permit us to construct implicit solvers using each of these spatial discretizations, wherein a chief difficulty is how to handle the higher derivatives in time. The end result is that the multiderivative time integrator allows us to obtain high-order accuracy in time while keeping the number of implicit stages at a minimum. We show numerical results validating and comparing methods.

Higher-Order Exponential Integrators for Quasi-Linear Parabolic Problems. Part II: Convergence

In this work, the convergence analysis of explicit exponential time integrators based on general linear methods for quasi-linear parabolic initial boundary value problems is pursued. Compared to other types of exponential integrators encountering rather severe order reductions, in general, the considered class of exponential general linear methods provides the possibility of constructing schemes that retain higher-order accuracy in time when applied to quasi-linear parabolic problems. In view of practical applications, the case of variable time step sizes is incorporated. The convergence analysis is based upon two fundamental ingredients. The needed stability bounds, obtained under mild restrictions on the ratios of subsequent time step sizes, have been deduced in the recent work [C. Gonzalez and M. Thalhammer, SIAM J. Numer. Anal., 53 (2015), pp. 701--719]. The core of the present work is devoted to the derivation of suitable local and global error representations. In conjunction with the stability bound...

Higher-order in time “quasi-unconditionally stable” ADI solvers for the compressible Navier–Stokes equations in 2D and 3D curvilinear domains

This paper introduces alternating-direction implicit (ADI) solvers of higher order of time-accuracy (orders two to six) for the compressible Navier-Stokes equations in two- and three-dimensional curvilinear domains. The higher-order accuracy in time results from 1) An application of the backward differentiation formulae time-stepping algorithm (BDF) in conjunction with 2) A BDF-like extrapolation technique for certain components of the nonlinear terms (which makes use of nonlinear solves unnecessary), as well as 3) A novel application of the Douglas-Gunn splitting (which greatly facilitates handling of boundary conditions while preserving higher-order accuracy in time). As suggested by our theoretical analysis of the algorithms for a variety of special cases, an extensive set of numerical experiments clearly indicate that all of the BDF-based ADI algorithms proposed in this paper are "quasi-unconditionally stable" in the following sense: each algorithm is stable for all couples $(h,\Delta t)$ of spatial and temporal mesh sizes in a problem-dependent rectangular neighborhood of the form $(0,M_h)\times (0,M_t)$. In other words, for each fixed value of $\Delta t$ below a certain threshold, the Navier-Stokes solvers presented in this paper are stable for arbitrarily small spatial mesh-sizes. The second order formulation has further been rigorously shown to be unconditionally stable for linear hyperbolic and parabolic equations in two-dimensional space. Although implicit ADI solvers for the Navier-Stokes equations with nominal second-order of temporal accuracy have been proposed in the past, the algorithms presented in this paper are the first ADI-based Navier-Stokes solvers for which second order or better accuracy has been verified in practice under non-trivial (non-periodic) boundary conditions.

Generalized Square Conservative Multistep Finite Difference Scheme Incorporating Historical Observations

In this paper,a multistep finite difference scheme has been proposed,whose coefficients are determined taking into consideration compatibility and generalized quadratic conservation,as well as incorporating historical observation data.The schemes have three advantages:high-order accuracy in time,generalized square conservation,and smart use of historical observations.Numerical tests based on the one-dimensional linear advection equations suggest that reasonable consideration of accuracy,square conservation,and inclusion of historical observations is critical for good performance of a finite difference scheme.

High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II)

Abstract In this paper, we first establish a high-order numerical algorithm for α-th (0 < α < 1) order Caputo derivative of a given function f(t), where the convergence rate is (4 − α)-th order. Then by using this new formula, an improved difference scheme with high order accuracy in time to solve Caputo-type fractional advection-diffusion equation with Dirichlet boundary conditions is constructed. Finally, numerical examples are carried out to confirm the efficiency of the constructed algorithm.

Higher-Order Exponential Integrators for Quasi-Linear Parabolic Problems. Part I: Stability

Explicit exponential integrators based on general linear methods are studied for the time discretization of quasi-linear parabolic initial-boundary value problems. Compared to other exponential integrators encountering rather severe order reductions, in general, the considered class of exponential general linear methods provides the possibility to construct schemes that retain higher-order accuracy in time when applied to quasi-linear parabolic problems. Employing an abstract framework, the considered problems take the form of initial value problems on Banach spaces: $u'(t) = Q(u(t)) u(t), t \in (0,T), u(0)$ given. A fundamental requirement for the stability and error analysis is that the domains of the defining sectorial operators $Q(v): D = D(Q(v)) \to X$ are independent of $v \in V \subset X$. The scope of applications in particular includes quasi-linear parabolic evolution equations subject to Dirichlet boundary conditions. The work is divided into two parts. In Part I, stability bounds in the norms of certain intermediate spaces between the domain $D$ and the underlying Banach space $X$ are deduced. In view of practical applications, the stability estimates are stated for variable time stepsizes, under mild restrictions on the ratios of subsequent stepsizes. The stability results provide a basic ingredient for the convergence analysis given in Part II.

Gauss–Runge–Kutta time discretization of wave equations on evolving surfaces

This paper is concerned with an error analysis for a full discretization of the linear wave equation on a moving surface. The equation is discretized in space by the evolving surface finite element method. Discretization in time is done by Gauβ---Runge---Kutta (GRK) methods, aiming for higher-order accuracy in time and unconditional stability of the fully discrete scheme. The latter is established in the natural time-dependent norms by using the algebraic stability and the coercivity property of the GRK methods together with the properties of the spatial semi-discretization. Under sufficient regularity conditions, optimal-order error estimates for this class of fully discrete methods are shown. Numerical experiments are presented to confirm some of the theoretical results.

Arbitrarily high order Convected Scheme solution of the Vlasov–Poisson system

The Convected Scheme (CS) is a ‘forward-trajectory’ semi-Lagrangian method for solution of transport equations, which has been most often applied to the kinetic description of plasmas and rarefied neutral gases. In its simplest form, the CS propagates the solution forward in time by advecting the so-called ‘moving cells’ along their characteristic trajectories, and by remapping them on the mesh at the end of the time step. The CS is conservative, positivity preserving, simple to implement, and it is not subject to time step restriction to maintain stability. Recently (Güçlü and Hitchon, 2012 [1]) a new methodology was introduced for reducing numerical diffusion, based on a modified equation analysis: the remapping error was compensated by applying small corrections to the final position of the moving cells prior to remapping. While the spatial accuracy was increased from 2nd to 4th order, the new scheme retained the important properties of the original method, and was shown to be extremely simple and efficient for constant advection problems.Here the CS is applied to the solution of the Vlasov–Poisson system, which describes the evolution of the velocity distribution function of a collection of charged particles subject to reciprocal Coulomb interactions. The Vlasov equation is split into two constant advection equations, one in configuration space and one in velocity space, and high order time accuracy is achieved by proper composition of the operators. The splitting procedure enables us to use the constant advection solver, which we extend to arbitrarily high order of accuracy in time and space: a new improved procedure is given, which makes the calculation of the corrections straightforward. Focusing on periodic domains, we describe a spectrally accurate scheme based on the fast Fourier transform; the proposed implementation is strictly conservative and positivity preserving. The ability to correctly reproduce the system dynamics, as well as resolving small-scale features in the solution, is shown in classical 1D–1V test cases, both in the linear and the non-linear regimes.

A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications

In the present work, first, a new fractional numerical differentiation formula (called the L1-2 formula) to approximate the Caputo fractional derivative of order α (0<α<1) is developed. It is established by means of the quadratic interpolation approximation using three points (tj−2,f(tj−2)),(tj−1,f(tj−1)) and (tj,f(tj)) for the integrand f(t) on each small interval [tj−1,tj] (j⩾2), while the linear interpolation approximation is applied on the first small interval [t0,t1]. As a result, the new formula can be formally viewed as a modification of the classical L1 formula, which is obtained by the piecewise linear approximation for f(t). Both the computational efficiency and numerical accuracy of the new formula are superior to that of the L1 formula. The coefficients and truncation errors of this formula are discussed in detail. Two test examples show the numerical accuracy of L1-2 formula. Second, by the new formula, two improved finite difference schemes with high order accuracy in time for solving the time-fractional sub-diffusion equations on a bounded spatial domain and on an unbounded spatial domain are constructed, respectively. In addition, the application of the new formula into solving fractional ordinary differential equations is also presented. Several numerical examples are computed. The comparison with the corresponding results of finite difference methods by the L1 formula demonstrates that the new L1-2 formula is much more effective and more accurate than the L1 formula when solving time-fractional differential equations numerically.

High-order unstructured Lagrangian one-step WENO finite volume schemes for non-conservative hyperbolic systems: Applications to compressible multi-phase flows

In this article we present the first better than second order accurate unstructured Lagrangian-type one-step WENO finite volume scheme for the solution of hyperbolic partial differential equations with non-conservative products. The method achieves high order of accuracy in space together with essentially non-oscillatory behavior using a non-linear WENO reconstruction operator on unstructured triangular meshes. High order accuracy in time is obtained via a local Lagrangian space–time Galerkin predictor method that evolves the spatial reconstruction polynomials in time within each element. The final one-step finite volume scheme is derived by integration over a moving space–time control volume, where the non-conservative products are treated by a path-conservative approach that defines the jump terms on the element boundaries. The entire method is formulated as an Arbitrary-Lagrangian–Eulerian (ALE) method, where the mesh velocity can be chosen independently of the fluid velocity.The new scheme is applied to the full seven-equation Baer–Nunziato model of compressible multi-phase flows with relaxation source terms in two space dimensions. The use of a Lagrangian approach allows an excellent resolution of the solid contact and the resolution of jumps in the volume fraction. The high order of accuracy of the scheme in space and time is confirmed via a numerical convergence study. Finally, the proposed method is also applied to a reduced version of the compressible Baer–Nunziato model for the simulation of free surface water waves in moving domains. In particular, the phenomenon of sloshing is studied in a moving water tank and comparisons with experimental data are provided.

Holberg's optimisation for high-order compact finite difference staggered schemes

Two optimised high-order compact finite difference (FD) staggered schemes are presented in this communication. Following Holberg's optimisation strategy, the least squares problem to minimising the group velocity (MGV) error, for the fourth- and sixth-order pentadiagonal schemes, is formulated. For a fixed level of group velocity accuracy, the optimised spectrum of wave number and the optimised coefficients for the schemes, are analytically evaluated. The spectral accuracy of these schemes has been verified by several comparisons with the FD staggered schemes obtained following Kim and Lee's (1996) optimisation procedure. Fewer group and phase velocity errors, greater resolution in terms of absolute error and resolving efficiency have been achieved by the optimised schemes proposed. High-order accuracy in time is obtained by marching the solution with an optimised Runge–Kutta scheme. Next, the comparison in terms of the number of grid points per wavelength required to achieve a standard accuracy for distances expressed in terms of the number of wavelengths travelled is presented. Numerical results from benchmark tests for the one-dimensional shallow water equations are presented.

A high order moving boundary treatment for compressible inviscid flows

We develop a high order numerical boundary condition for compressible inviscid flows involving complex moving geometries. It is based on finite difference methods on fixed Cartesian meshes which pose a challenge that the moving boundaries intersect the grid lines in an arbitrary fashion. Our method is an extension of the so-called inverse Lax–Wendroff procedure proposed in [17] for conservation laws in static geometries. This procedure helps us obtain normal spatial derivatives at inflow boundaries from Lagrangian time derivatives and tangential derivatives by repeated use of the Euler equations. Together with high order extrapolation at outflow boundaries, we can impose accurate values of ghost points near the boundaries by a Taylor expansion. To maintain high order accuracy in time, we need some special time matching technique at the two intermediate Runge–Kutta stages. Numerical examples in one and two dimensions show that our boundary treatment is high order accurate for problems with smooth solutions. Our method also performs well for problems involving interactions between shocks and moving rigid bodies.

Consistent discretization for simulations of flows with moving generalized curvilinear coordinates

AbstractWe develop a consistent discretization of conservative momentum and scalar transport for the numerical simulation of flow using a generalized moving curvilinear coordinate system. The formulation guarantees consistency between the discrete transport equation and the discrete mass conservation equation due to grid motion. This enables simulation of conservative transport using generalized curvilinear grids that move arbitrarily in three dimensions while maintaining the desired properties of the discrete transport equation on a stationary grid, such as constancy, conservation, and monotonicity. In addition to guaranteeing consistency for momentum and scalar transport, the formulation ensures geometric conservation and maintains the desired high‐order time accuracy of the discretization on a moving grid. Through numerical examples we show that, when the computation is carried out on a moving grid, consistency between the discretized scalar advection equation and the discretized equation for flow mass conservation due to grid motion is required in order to obtain stable and accurate results. We also demonstrate that significant errors can result when non‐consistent discretizations are employed. Copyright © 2009 John Wiley &amp; Sons, Ltd.

Characteristic-based finite volume scheme for 1D Euler equations

In this paper, a high-order finite-volume scheme is presented for the onedimensional scalar and inviscid Euler conservation laws. The Simpson’s quadrature rule is used to achieve high-order accuracy in time. To get the point value of the Simpson,s quadrature, the characteristic theory is used to obtain the positions of the grid points at each sub-time stage along the characteristic curves, and the third-order and fifth-order central weighted essentially non-oscillatory (CWENO) reconstruction is adopted to estimate the cell point values. Several standard one-dimensional examples are used to verify the high-order accuracy, convergence and capability of capturing shock.

Implicit high-order unconditionally stable complex envelope algorithm for solving the time-dependent Maxwell's equations

Based on the Padé approximation and multistep method, we propose an implicit high-order unconditionally stable complex envelope algorithm to solve the time-dependent Maxwell's equations. Unconditional numerical stability can be achieved simultaneously with a high-order accuracy in time. As we adopt the complex envelope Maxwell's equations, numerical dispersion and dissipation are very small even at comparatively large time steps. To verify the capability of our algorithm, we compare the results of the proposed method with the exact solutions.