High-order Accuracy In Time Research Articles

Energy dissipation law of the variable time-step fractional BDF2 scheme for the time fractional molecular beam epitaxial model

In this work, we take a consideration of the time fractional molecular beam epitaxial(MBE) models. The variable time-step BDF2 methods are proposed for time fractional MBE models in order to obtain the high-order accuracy in time since the low regularity occurring in the initial state. By virtue of discrete gradient structures, energy dissipation laws of the schemes for time fractional MBE model with and without slope selection are proved, respectively. Besides, the unique solvabilities of two schemes are also guaranteed. The discrete modified energy and the corresponding energy dissipation law are asymptotically compatible with the associated discrete energy and the energy dissipation law of the variable time-step BDF2 method for the classical MBE model. Numerical experiments with the time adaptive strategies are provided to verify the effectiveness of the schemes.

A Compact Split-step Finite Difference Method for Solving the Nonlinear Schrödinger Equation

The nonlinear Schrödinger equation arises from quantum mechanics and is extensively used in many fields of science and engineering. Thus, it is important to construct the high-order and stable numerical scheme of the Schrödinger equation. To solve the high-order and stable numerical solution of the nonlinear Schrödinger equation, the compact split-step finite difference method and the local one-dimensional method are combined in this paper. To attain high-order accuracy in time and space, the 4-order compact finite difference in space discretization is combined with the L-stable Simpson method in time discretization. Therefore, a scheme with 4-order accuracy in space and 3-order accuracy in time is obtained, and the stability of the scheme is analyzed. Finally, numerical results manifest that the devised scheme can supply accurate and stable results to the nonlinear Schrödinger equation.

Exponential DG methods for Vlasov equations

In this work, an exponential Discontinuous Galerkin (DG) method is proposed to solve numerically Vlasov type equations. The DG method is used for space discretization which is combined exponential Lawson Runge-Kutta method for time discretization to get high order accuracy in time and space. In addition to get high order accuracy in time, the use of Lawson methods enables to overcome the stringent condition on the time step induced by the linear part of the system. Moreover, it can be proved that a discrete Poisson equation is preserved. Numerical results on Vlasov-Poisson and Vlasov Maxwell equations are presented to illustrate the good behavior of the exponential DG method.

Numerical analysis of a space-time continuous Galerkin method for the nonlinear Sobolev equation

In this work, we shall use space-time continuous Galerkin (STCG) method to analyze the nonlinear Sobolev equation. This method is different from the conventional finite element methods, that is, it not only uses finite element to discrete spatial variables but also uses finite element to discrete temporal variable. Therefore it is easier to obtain high order accuracy in time and space than the conventional finite element methods and has the good stability. Also, it does not need time step size and space mesh parameter to satisfy the stability conditions. In this paper, we show a detailed analysis including the proof of the existence and uniqueness of the numerical solution and the a priori error estimate. Especially, the analysis method given here can be similarly applied to the more general nonlinear problems. At last, the numerical examples reveal that the iterative method is more efficient than the linearized method for the nonlinear problems with STCG method. Also, the numerical experiments show that the STCG method is more efficient than the space-time discontinuous Galerkin (STDG) method in the practical calculations.

A Kronecker product linear-cost solver for the high-order generalized-α method for multi-dimensional hyperbolic systems

The main result of the work is the generalization of the concept of the variational splitting from [3] to the higher-order generalized-α schemes proposed in [5] in the context of hyperbolic and elastic wave propagation problems. It is possible to factorize the matrix of the linear system of equations (M+ηK) in the linear computational cost. Here M is the mass matrix, and K is the stiffness matrix. We discretize using the isogeometric finite element method. Following the idea proposed in [3], factorizing the matrix (M+ηK), we obtained an additional error of the order of η2. This paper introduces a technique to couple the possibility of fast factorization of (M+ηK) with higher-order generalized-α schemes proposed in [5] for hyperbolic and elastic wave propagation problems. Thus, we introduce a variationally separable splitting technique for the high-order accuracy generalized-α method. We use tensor-product meshes to develop the splitting method, which results in the linear cost for multi-dimensional problems. We consider standard C0 finite elements as well as smoother isogeometric analysis for spatial discretization. The direction splitting method requires the mass and the stiffness matrices to have the Kronecker product structure. For other problems, the corresponding Kronecker product solver can be employed as a preconditioner as long as a Kronecker product can sufficiently well approximate the problem matrix. We also study the spectrum of the amplification matrix to establish the unconditional stability of the method. We use various examples to demonstrate the performance of the method and the optimal approximation accuracy. In numerical tests, we compute the L2 and H1 norms to show the optimal convergence of the discrete method in space and high-order accuracy in time.

A new family of semi-implicit Finite Volume/Virtual Element methods for incompressible flows on unstructured meshes

We introduce a new family of high order accurate semi-implicit schemes for the solution of nonlinear time-dependent systems of partial differential equations (PDE) on unstructured polygonal meshes. The time discretization is based on a splitting between explicit and implicit terms that may arise either from the multi-scale nature of the governing equations, which involve both slow and fast scales, or in the context of projection methods, where the numerical solution is projected onto the physically meaningful solution manifold. We propose to use a high order finite volume (FV) scheme for the explicit terms, hence ensuring conservation property and robustness across shock waves, while the virtual element method (VEM) is employed to deal with the discretization of the implicit terms, which typically requires an elliptic problem to be solved. The numerical solution is then transferred via suitable L2 projection operators from the FV to the VEM solution space and vice-versa. High order time accuracy is then achieved using the semi-implicit IMEX Runge–Kutta schemes, and the novel schemes are proven to be asymptotic preserving (AP) and well-balanced (WB). As representative models, we choose the shallow water equations (SWE), thus handling multiple time scales characterized by a different Froude number, and the incompressible Navier–Stokes equations (INS), which are solved at the aid of a projection method to satisfy the solenoidal constraint of the velocity field. Furthermore, an implicit discretization for the viscous terms is also devised for the INS model, which is based on the VEM technique. Consequently, the CFL-type stability condition on the maximum admissible time step is based only on the fluid velocity and not on the celerity nor on the viscous eigenvalues. A large suite of test cases demonstrates the accuracy and the capabilities of the new family of schemes to solve relevant benchmarks in the field of incompressible fluids.

High order asymptotic preserving finite difference WENO schemes with constrained transport for MHD equations in all sonic Mach numbers

In this paper, a high-order semi-implicit (SI) asymptotic preserving (AP) and divergence-free finite difference weighted essentially nonoscillatory (WENO) scheme is proposed for magnetohydrodynamic (MHD) equations. We consider the sonic Mach number ε ranging from 0 to O(1). High-order accuracy in time is obtained by SI implicit-explicit Runge–Kutta (IMEX-RK) time discretization. High-order accuracy in space is achieved by finite difference WENO schemes with characteristic-wise reconstructions. A constrained transport method is applied to maintain a discrete divergence-free condition. We formally prove that the scheme is AP. Asymptotic accuracy (AA) in the incompressible MHD limit is obtained if the implicit part of the SI IMEX-RK scheme is stiffly accurate. Numerical experiments are provided to validate the AP, AA, and divergence-free properties of our proposed approach. Besides, the scheme can well capture discontinuities such as shocks in an essentially non-oscillatory fashion in the compressible regime, while it is also a good incompressible solver with uniform large-time step conditions in the low sonic Mach limit.

On convergence of difference schemes of high accuracy for one pseudo-parabolic Sobolev type equation

Difference schemes of the finite difference method and the finite element method of high-order accuracy in time and space are proposed and investigated for a pseudo-parabolic Sobolev type equation. The order of accuracy in space is improved in two ways using the finite difference method and the finite element method. The order of accuracy of the scheme in time is improved by a special discretization of the time variable. The corresponding a priori estimates are determined and, on their basis, the accuracy estimates of the proposed difference schemes are obtained with sufficient smoothness of the solution to the original differential problem. Algorithms for the implementation of the constructed difference schemes are proposed.

Deneb: An open-source high-performance multi-physical flow solver based on high-order DRM-DG method

High-order methods are being recognized as powerful tools for handling scale-resolving simulations over complex geometry. However, several obstacles still block their complete applications to practical engineering problems: a compromise between accuracy and efficiency on mixed-curved meshes, inherent vulnerability to numerical oscillations, and lack of open-source high-performance solvers for researchers. To address these issues, we present Deneb, an open-source high-order accurate numerical solver that enables high-performance scale-resolving simulations on PDE-based flow systems. Deneb uses the physical domain-based modal discontinuous Galerkin (DG) method; thus, it can provide an arbitrary high-order accurate solution on mixed-curved meshes and has the potential for handling polyhedral meshes as well. The direct reconstruction method (DRM) efficiently executes the numerical integration of DG volume and surface integrals without accuracy loss on non-affine elements where mapping functions are high-degree. The resulting DRM-DG method eliminates the severe cost of a quadrature-based approach on mixed-curved meshes. Deneb offers explicit and implicit Runge–Kutta methods as well to achieve high-order accuracy in time. In addition, Krylov subspace methods and preconditioners are available for high-performance linear system solving in parallel. Deneb possesses a strong capability to resolve multi-physical shocks without numerical instabilities with the aid of multi-dimensional limiting and artificial viscosity methods. In particular, the hierarchical multi-dimensional limiting process enables efficient computations of supersonic flows without time-step restriction. The current release of Deneb covers the simulations of hypersonic equilibrium and magneto-hydrodynamic flows as well as compressible Navier–Stokes equations, but it has the potential to solve any PDE-based multi-physical flow systems. Several benchmark problems are presented to highlight Deneb's capability to perform scale-resolving and multi-physical flow simulations. A scalability test is also presented to verify the scaling characteristics of Deneb for high-performance computing. Program summaryProgram title: DenebCPC library link to program files:https://doi.org/10.17632/723n5r797n.1Developer's repository link:https://github.com/HojunYouKr/DenebLicensing provisions: BSD-3-ClauseProgramming language: C++17Nature of problem: The physical domain-based modal DG method can achieve the expected order of accuracy with the optimal number of polynomial bases even on non-affine elements. However, the DG method becomes significantly expensive on high-order curved elements when using quadrature rules, blocking its applicability to practical engineering problems. The numerical integration should be much more efficient without compromising accuracy. In addition, the less diffusive nature of high-order methods makes them susceptible to producing spurious numerical oscillations near flow discontinuities, potentially leading to numerical instabilities. Thus, an accurate and robust shock-capturing method is essential to simulate multi-physical flows under compressible regimes. Finally, the solver needs high scalability to perform large-scale computations in parallel.Solution method: DRM is applied to the DG volume and surface integrals to perform efficient numerical integration on non-affine elements without accuracy loss. The resulting method, DRM-DG, provides arbitrary high-order accurate solutions to various PDE-based flow problems on mixed-curved meshes. The solution is also high-order accurate in time due to high-order explicit and implicit Runge–Kutta methods implemented. The external library enables high-performance linear system solving with various preconditioners in parallel. Both multi-dimensional limiting and artificial viscosity methods suppress unwanted subcell oscillations across physical discontinuities. In particular, the limiting methods simulate complex supersonic flows efficiently without time-step restriction. The solver is highly scalable on parallel computing with the aid of non-blocking communications and latency hiding.

An all Froude high order IMEX scheme for the shallow water equations on unstructured Voronoi meshes

We propose a novel numerical method for the solution of the shallow water equations in different regimes of the Froude number making use of general polygonal meshes. The fluxes of the governing equations are split such that advection and acoustic-gravity sub-systems are derived, hence separating slow and fast phenomena. This splitting allows the nonlinear convective fluxes to be discretized explicitly in time, while retaining an implicit time marching for the acoustic-gravity terms. Consequently, the novel schemes are particularly well suited in the low Froude limit of the model, since no numerical viscosity is added in the implicit solver. Besides, stability follows from a milder CFL condition which is based only on the advection speed and not on the celerity. High order time accuracy is achieved using the family of semi-implicit IMEX Runge-Kutta schemes, while high order in space is granted relying on two discretizations: (i) a cell-centered finite volume (FV) scheme for the nonlinear convective contribution on the polygonal cells; (ii) a staggered discontinuous Galerkin (DG) scheme for the solution of the linear system associated to the implicit discretization of the pressure sub-system. Therefore, three different meshes are used, namely a polygonal Voronoi mesh, a triangular subgrid and a staggered quadrilateral subgrid. The novel schemes are proved to be Asymptotic Preserving (AP), hence a consistent discretization of the limit model is retrieved for vanishing Froude numbers, which is given by the so-called “lake at rest” equations. Furthermore, the novel methods are well-balanced by construction, and this property is also demonstrated. Accuracy and robustness are then validated against a set of benchmark test cases with Froude numbers ranging in the interval Fr≈[10−6;5], hence showing that multiple time scales can be handled by the novel methods.

Maximum Principle Preserving Space and Time Flux Limiting for Diagonally Implicit Runge–Kutta Discretizations of Scalar Convection-diffusion Equations

We provide a framework for high-order discretizations of nonlinear scalar convection-diffusion equations that satisfy a discrete maximum principle. The resulting schemes can have arbitrarily high order accuracy in time and space, and can be stable and maximum-principle-preserving (MPP) with no step size restriction. The schemes are based on a two-tiered limiting strategy, starting with a high-order limiter-based method that may have small oscillations or maximum-principle violations, followed by an additional limiting step that removes these violations while preserving high order accuracy. The desirable properties of the resulting schemes are demonstrated through several numerical examples.

High Order Semi-implicit WENO Schemes for All-Mach Full Euler System of Gas Dynamics

In this paper, we propose high order semi-implicit schemes for the all Mach full Euler equations of gas dynamics. Material waves are treated explicitly, while acoustic waves are treated implicitly, thus avoiding severe CFL restrictions for low Mach flows. High order accuracy in time is obtained by semi-implicit temporal integrator based on the IMEX Runge-Kutta (IMEX-RK) framework. High order in space is achieved by finite difference WENO schemes with characteristic-wise reconstructions adapted to the semi-implicit IMEX-RK time discretization. The schemes are proven to be asymptotic preserving and asymptotically accurate as the Mach number vanishes. Besides, they can well capture discontinuous solutions in the compressible regime, especially for two dimensional Riemann problems. Numerical tests in one and two space dimensions will illustrate the effectiveness of the proposed schemes.

A recursive system-free single-step temporal discretization method for finite difference methods

Single-stage or single-step high-order temporal discretizations of partial differential equations (PDEs) have shown great promise in delivering high-order accuracy in time with efficient use of computational resources. There has been much success in developing such methods for finite volume method (FVM) discretizations of PDEs. The Picard Integral formulation (PIF) has recently made such single-stage temporal methods accessible for finite difference method (FDM) discretizations. PIF methods rely on the so-called Lax-Wendroff procedures to tightly couple spatial and temporal derivatives through the governing PDE system to construct high-order Taylor series expansions in time. Going to higher than third order in time requires the calculation of Jacobian-like derivative tensor-vector contractions of an increasingly larger degree, greatly adding to the complexity of such schemes. To that end, we present in this paper a method for calculating these tensor contractions through a recursive application of a discrete Jacobian operator that readily and efficiently computes the needed contractions entirely agnostic of the system of partial differential equations (PDEs) being solved.

Fast numerical schemes for nonlinear space-fractional multidelay reaction-diffusion equations by implicit integration factor methods

In this article, a fast and efficient numerical method is constructed for solving nonlinear space-fractional multidelay reaction-diffusion equations. Firstly, we spatially discretize the equation using the weighted and shifted Grünwald–Letnikov difference (WSGD) formula. As a result, a nonlinear system of ordinary differential equations (ODEs) is obtained. Then, based on the fact that the implicit integration factor (IIF) method is an effective time-stepping scheme with good stability properties, we develop a multidelay IIF (MIIF) method to deal with the resulting ODE system. Compared with traditional numerical schemes, such as the backward differential formula (BDF) and the Crank–Nicolson scheme (CN), the proposed MIIF scheme has two main advantages: (1) MIIF can achieve high-order accuracy in time; (2) the temporal errors are smaller and the convergence orders observed with MIIF are more regular. In addition, in order to overcome the computational challenges, we propose some Krylov subspace methods to calculate the actions of the Toeplitz matrix exponentials in MIIF. Finally, numerical examples are presented to confirm the accuracy of the MIIF scheme and to demonstrate the considerable computational advantages of the proposed fast solving algorithms.

Conservative semi-Lagrangian schemes for kinetic equations Part II: Applications

In this paper, we present a new class of conservative semi-Lagrangian schemes for kinetic equations. They are based on the conservative reconstruction technique introduced in [1]. The methods are high order accurate both in space and time. Because of the semi-Lagrangian nature, the time step is not restricted by a CFL-type condition. Applications are presented to the Vlasov-Poisson system and the BGK model of rarefied gas dynamics. In the first case, operator splitting is adopted to obtain high order accuracy in time, and a conservative reconstruction that preserves the maximum and minimum of the function is used. For initially positive solutions, in particular, this guarantees exact preservation of the L1-norm. Conservative schemes for the BGK model are constructed by coupling the conservative reconstruction with a conservative treatment of the collision term. High order in time is obtained by either Runge-Kutta or BDF time discretization of the equation along characteristics. Because of L-stability and exact conservation, the resulting scheme for the BGK model is asymptotic preserving for the underlying fluid dynamic limit. Several test cases in one and two space dimensions confirm the accuracy and robustness of the methods, and the AP property of the schemes when applied to the BGK model.

Generalized Petrov-Galerkin time finite element weighted residual methodology for designing high-order unconditionally stable algorithms with controllable numerical dissipation

In this paper, a novel stabilized time-weighted residual methodology under the umbrella of Petrov-Galerkin time finite element formulation is developed to design a generalized computational framework, which permits unconditionally stable, high-order time accuracy, and features with controllable numerical dissipation for solving transient first-order systems. Various unconditionally stable (A/L-stable) algorithms can be readily obtained in the proposed framework having not only high-order accuracy but also controllable numerical dissipation in the high frequency. Quadratic and cubic basis functions are utilized to illustrate the specific design process of the proposed method, which consequently ends up with numerous third-/fifth-order time accurate algorithms, QUAD3(γ,ρ∞) and CUBE5(γ,ρ∞), with controllable numerical dissipation. Comparing with the well-known implicit Runge-Kutta (RK) family of algorithms, these newly developed QUAD3(γ,ρ∞) and CUBE5(γ,ρ∞) can (a) recover the Radau IIA3/RK3 and Radau IIA5/RK5 schemes by certain selection of algorithmic parameters (γ,ρ∞); (b) obtain numerous new algorithms with improved solution accuracy that is superior to the RK family of algorithms, and (c) has similar computational efficiency as that of the implicit RK family of algorithms. Several single/multi-degree of freedom (SDOF and MDOF) problems are investigated to validate the proposed developments. In addition, it is worth noting that the proposed methodology can also use high-order (not limited to the quadratic and cubic) basis functions to design more advanced schemes and can be integrated with high-order spatial discretization methods in the solution of space-time PDEs, such as isogeometric methods, SEM, Discontinuous Galerkin (DGM), p-versions FEM, etc.

Projection methods for the incompressible Navier-Stokes equations

The incompressible Navier-Stokes equations (INSE) are the basic governing equations of fluid dynamics, and their numerical solutions are of great significance. In this review paper, we first recollect some classical projection methods and their relatives in the past 50 years and then fully explain the recent fourth-order projection method called GePUP [Zhang Q <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1007/s10915-015-0122-4">2016 <i>J. Sci. Comput.</i> <b>67</b> 1134</ext-link>]. Based on a generic projection operator and the UPPE formulation of the INSE [Liu J G, Liu J, Pego R L <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1016/j.jcp.2011.08.011">2007 <i>Comm. Pure Appl. Math</i>. <b>60</b> 1443</ext-link>], we derive the GePUP equations, which retain the advantage of UPPE that the velocity divergence is governed by a heat equation and is thus well under control. In comparison with UPPE, the GePUP formulation is advantageous in three aspects: (1) its derivation depends on none of the properties of the Leray-Helmholtz projection; (2) the evolutionary velocity can be divergent, thus it is directly applicable to numerical calculations with nonzero velocity divergence; (3) the Leray-Helmholtz projection does appear on the right-hand sides of the governing equations, thus making it transparent to analyze the accuracy and stability issues raised by numerically approximating the Leray-Helmholtz projection. As the most appealing feature of GePUP, temporal integration and spatial discretization are completely decoupled and can be treated as black boxes, so that the user can choose his favorite methods for the two parts to form his own GePUP method. In particular, high-order accuracy in time can be easily obtained since no internal details of the ODE solver are needed. The flexibility in time makes the GePUP method applicable to both low-Reynolds-number flow and high-Reynolds-number flow. The flexibility in space makes the GePUP method applicable to both rectangular boxes and irregular domains. The numerical results and elementary analysis show that the fourth-order GePUP method may be much more accurate and efficient than classical second-order projection methods by many orders of magnitude.

A well-balanced ADER discontinuous Galerkin method based on differential transformation procedure for shallow water equations

This article develops a new discontinuous Galerkin (DG) method on structured meshes for solving shallow water equations. The method here applies the one-stage ADER (Arbitrary DERivatives in time and space) approach for the temporal discretization and employs the differential transformation procedure to express spatiotemporal expansion coefficients of the solution through low order spatial expansion coefficients recursively. Numerical fluxes using the hydrostatic reconstruction together with a simple source term discretization results in a well-balanced method accordingly. Compared with the Runge-Kutta DG (RKDG) methods, the proposed method needs less computer memory storage due to no intermediate stages. In comparison with traditional ADER schemes, this method avoids solving the generalized Riemann problems at inter-cells. The differential transformation procedure used here is cheaper than the Cauchy-Kowalewski procedure in traditional ADER schemes. In summary, this method is one-step, one-stage, fully-discrete, and easily achieves arbitrary high order accuracy in time and space. Theoretically as well as numerically, the proposed method is verified to be well-balanced. Numerical results illustrate the high order accuracy as well as good resolutions for discontinuous solutions. Moreover, this method is more efficient than the Runge-Kutta DG (RKDG) method.

Kernel Based High Order “Explicit” Unconditionally Stable Scheme for Nonlinear Degenerate Advection-Diffusion Equations

In this paper, we present a novel numerical scheme for solving a class of nonlinear degenerate parabolic equations with non-smooth solutions. The proposed method relies on a special kernel based formulation of the solutions found in our early work on the method of lines transpose and successive convolution. In such a framework, a high order weighted essentially non-oscillatory methodology and a nonlinear filter are further employed to avoid spurious oscillations. High order accuracy in time is realized by using the high order explicit strong-stability-preserving (SSP) Runge-Kutta method. Moreover, theoretical investigations of the kernel based formulation combined with an explicit SSP method indicate that the combined scheme is unconditionally stable and up to third order accuracy. Evaluation of the kernel based approach is done with a fast $${\mathcal {O}}(N)$$ summation algorithm. The new method allows for much larger time step evolution compared with other explicit schemes with the same order accuracy, leading to remarkable computational savings.

High order accurate schemes for Euler and Navier–Stokes equations on staggered Cartesian grids

We present here a new class of staggered schemes for solving the compressible 1D Euler equations in internal energy formulation on uniform grids. The schemes are applicable to arbitrary equation of states and can be extended to high order of accuracy in both time and space on smooth flows. High order accuracy in time is reached thanks to Cauchy–Kovalevskaya procedure. Modifications on the initial schemes are performed to give sufficient conditions for stability on 1D wave equations. Results obtained for wave equations are extended to 1D Euler equations and then to 2D compressible Navier–Stokes equations using directional splitting methods. Results on the conservation of total energy are given, proper shock capturing is observed experimentally. Numerical results are provided up to 4th-order accuracy in 1D and 2D.