High order strong stability preserving (SSP) time discretizations are advantageous for use with spatial discretizations with nonlinear stability properties for the solution of hyperbolic PDEs. The search for high order strong stability time-stepping methods with large allowable strong stability time-step has been an active area of research over the last two decades. Recently, multiderivative time-stepping methods have been implemented with hyperbolic PDEs. In this work we describe sufficient conditions for a two-derivative multistage method to be SSP, and find some optimal SSP multistage two-derivative methods. While explicit SSP Runge--Kutta methods exist only up to fourth order, we show that this order barrier is broken for explicit multi-stage two-derivative methods by designing a three stage fifth order SSP method. These methods are tested on simple scalar PDEs to verify the order of convergence, and demonstrate the need for the SSP condition and the sharpness of the SSP time-step in many cases.
In this paper, we introduce a fourth-order accurate finite element method for incompressible variable density flow. The method is implicit in time and constructed with the Taylor series technique, and uses standard high-order Lagrange basis functions in space. Taylor series time-stepping relies on time derivative correction terms to achieve high-order accuracy. We provide detailed algorithms to approximate the time derivatives of the variable density Navier-Stokes equations. Numerical validations confirm a fourth-order accuracy for smooth problems. We also numerically illustrate that the Taylo…
Adaptive second-order Crank-Nicolson time-stepping methods using the recent scalar auxiliary variable (SAV) approach are developed for the time-fractional Molecular Beam Epitaxial models with Caputo's derivative. Based on the piecewise linear interpolation, the Caputo's fractional derivative is approximated by a novel second-order formula, which is naturally suitable for a general class of nonuniform meshes and essentially preserves the positive semi-definite property of integral kernel. The resulting Crank-Nicolson SAV time-stepping schemes are unconditional energy stable on nonuniform time m…
Mean-square stability and error analysis of implicit time-stepping schemes for linear parabolic SPDEs with multiplicative Wiener noise in the first derivative
Two fast L1 time-stepping methods, including the backward Euler and stabilized semi-implicit schemes, are suggested for the time-fractional Allen-Cahn equation with Caputo's derivative. The time mesh is refined near the initial time to resolve the intrinsically initial singularity of solution, and unequal time-steps are always incorporated into our approaches so that an adaptive time-stepping strategy can be used in long-time simulations. It is shown that the proposed schemes using the fast L1 formula preserve the discrete maximum principle. Sharp error estimates reflecting the time regularity…
In this paper, a non-uniform time-stepping convex-splitting numerical algorithm for solving the widely used time-fractional Cahn-Hilliard equation is introduced. The proposed numerical scheme employs the $L1^+$ formula for discretizing the time-fractional derivative and a second-order convex-splitting technique to deal with the non-linear term semi-implicitly. Then the pseudospectral method is utilized for spatial discretization. As a result, the fully discrete scheme has several advantages: second-order accurate in time, spectrally accurate in space, uniquely solvable, mass preserving, and un…
The aim of this paper is to develop and analyze high-order time stepping schemes for solving semilinear subdiffusion equations. We apply the $k$-step BDF convolution quadrature to discretize the time-fractional derivative with order $α\in (0,1)$, and modify the starting steps in order to achieve optimal convergence rate. This method has already been well-studied for the linear fractional evolution equations in Jin, Li and Zhou \cite{JinLiZhou:correction}, while the numerical analysis for the nonlinear problem is still missing in the literature. By splitting the nonlinear potential term into an…
The aim of this paper is to study the time stepping scheme for approximately solving the subdiffusion equation with a weakly singular source term. In this case, many popular time stepping schemes, including the correction of high-order BDF methods, may lose their high-order accuracy. To fill in this gap, in this paper, we develop a novel time stepping scheme, where the source term is regularized by using a $k$-fold integral-derivative and the equation is discretized by using a modified BDF2 convolution quadrature. We prove that the proposed time stepping scheme is second-order, even if the sou…
In this work, we propose a Crank-Nicolson-type scheme with variable steps for the time fractional Allen-Cahn equation. The proposed scheme is shown to be unconditionally stable (in a variational energy sense), and is maximum bound preserving. Interestingly, the discrete energy stability result obtained in this paper can recover the classical energy dissipation law when the fractional order $α\rightarrow 1.$ That is, our scheme can asymptotically preserve the energy dissipation law in the $α\rightarrow 1$ limit. This seems to be the first work on variable time-stepping scheme that can preserve …
Time-fractional parabolic equations with a Caputo time derivative are considered. For such equations, we explore and further develop the new methodology of the a-posteriori error estimation and adaptive time stepping proposed in [7]. We improve the earlier time stepping algorithm based on this theory, and specifically address its stable and efficient implementation in the context of high-order methods. The considered methods include an L1-2 method and continuous collocation methods of arbitrary order, for which adaptive temporal meshes are shown to yield optimal convergence rates in the presen…
In this article, a high-order time-stepping scheme based on the cubic interpolation formula is considered to approximate the generalized Caputo fractional derivative (GCFD). Convergence order for this scheme is $(4-α)$, where $α~(0<α<1)$ is the order of the GCFD. The local truncation error is also provided. Then, we adopt the developed scheme to establish a difference scheme for the solution of generalized fractional advection-diffusion equation with Dirichlet boundary conditions. Furthermore, we discuss about the stability and convergence of the difference scheme. Numerical examples are prese…
Sharp $H^1$-norm error estimates of two time-stepping schemes for reaction-subdiffusion problems
Henan University of Economics and Law（河南财经政法大学）；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；Beijing Computational Science Research Center（北京计算科学研究中心）；Wayne State University（韦恩州立大学）
In this work, we consider a time-fractional Allen-Cahn equation, where the conventional first order time derivative is replaced by a Caputo fractional derivative with order $α\in(0,1)$. First, the well-posedness and (limited) smoothing property are systematically analyzed, by using the maximal $L^p$ regularity of fractional evolution equations and the fractional Grönwall's inequality. We also show the maximum principle like their conventional local-in-time counterpart. Precisely, the time-fractional equation preserves the property that the solution only takes value between the wells of the dou…
An Efficient and Accurate Two-Stage Fourth-order Gas-kinetic Scheme for the Navier-Stokes Equations
Hong Kong University of Science and Technology（香港科技大学）；Tsinghua University（清华大学）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）
In this paper, an implicit time stepping meshless scheme is proposed to find the numerical solution of high-dimensional sine-Gordon equations (SGEs) by combining the high dimensional model representation (HDMR) and the Fourier hyperbolic cross (HC) approximation. To ensure the sparseness of the relevant coefficient matrices of the implicit time stepping scheme, the whole domain is first divided into a set of subdomains, and the relevant derivatives in high-dimension can be separately approximated by the Fourier HDMR-HC approximation in each subdomain. The proposed method allows for stable larg…
Finite differences and Runge-Kutta time stepping schemes used in Computational AeroAcoustics simulations are often optimized for low dispersion and dissipation (e.g. DRP or LDDRK schemes) when applied to linear problems in order to accurately simulate waves with the least computational cost. Here, the performance of optimized Runge-Kutta time stepping schemes for linear time-invariant problems with non-constant-amplitude oscillations is considered. This is in part motivated by the recent suggestion that optimized spatial derivatives perform poorly for growing and decaying waves, as their optim…
Based on the recent development of Jacobian-free Lax-Wendroff (LW) approaches for solving hyperbolic conservation laws [Zorio, Baeza and Mulet, Journal of Scientific Computing 71:246-273, 2017], [Carrillo and Parés, Journal of Scientific Computing 80:1832-1866, 2019], a novel collection of explicit Jacobian-free multistage multiderivative solvers for hyperbolic conservation laws is presented in this work. In contrast to Taylor time-integration methods, multiderivative RungeKutta (MDRK) techniques achieve higher-order of consistency not only through the excessive addition of higher temporal der…
Long-time stability of implicit-explicit Runge-Kutta methods for two-dimensional incompressible flows
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；Eastern Institute of Technology（宁波东方理工学院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
A fully implicit numerical scheme is established for solving the time fractional Swift-Hohenberg (TFSH) equation with a Caputo time derivative of order $α\in(0,1)$. The variable-step L1 formula and the finite difference method are employed for the time and the space discretizations, respectively. The unique solvability of the numerical scheme is proved by the Brouwer fixed-point theorem. With the help of the discrete convolution form of L1 formula, the time-stepping scheme is shown to preserve a discrete energy dissipation law which is asymptotically compatible with the classic energy law as $…
In this paper, we propose and analyze a time-stepping method for the time fractional Allen-Cahn equation. The key property of the proposed method is its unconditional stability for general meshes, including the graded mesh commonly used for this type of equations. The unconditional stability is proved through establishing a discrete nonlocal free energy dispassion law, which is also true for the continuous problem. The main idea used in the analysis is to split the time fractional derivative into two parts: a local part and a history part, which are discretized by the well known L1, L1-CN, and…
This paper develops three high-order accurate discontinuous Galerkin (DG) methods for the one-dimensional (1D) and two-dimensional (2D) nonlinear Dirac (NLD) equations with a general scalar self-interaction. They are the Runge-Kutta DG (RKDG) method and the DG methods with the one-stage fourth-order Lax-Wendroff type time discretizaiton (LWDG) and the two-stage fourth-order accurate time discretization (TSDG). The RKDG method uses the spatial DG approximation to discretize the NLD equations and then utilize the explicit multistage high-order Runge-Kutta time discretization for the first-order …
A time-stepping L1 scheme for subdiffusion equation with a Riemann--Liouville time-fractional derivative is developed and analyzed. This is the first paper to show that the L1 scheme for the model problem under consideration is second-order accurate (sharp error estimate) over nonuniform time-steps. The established convergence analysis is novel, innovative and concise. For completeness, the L1 scheme is combined with the standard Galerkin finite elements for the spatial discretization, which will then define a fully-discrete numerical scheme. The error analysis for this scheme is also investig…
In this work, we develop an efficient incomplete iterative scheme for the numerical solution of the subdiffusion model involving a Caputo derivative of order $α\in(0,1)$ in time. It is based on piecewise linear Galerkin finite element method in space and backward Euler convolution quadrature in time and solves one linear algebraic system inexactly by an iterative algorithm at each time step. We present theoretical results for both smooth and nonsmooth solutions, using novel weighted estimates of the time-stepping scheme. The analysis indicates that with the number of iterations at each time le…
We propose a new discretization method for PDEs on moving domains in the setting of unfitted finite element methods, which is provably higher-order accurate in space and time. In the considered setting, the physical domain that evolves essentially arbitrarily through a time-independent computational background domain, is represented by a level set function. For the time discretization, the application of standard time stepping schemes that are based on finite difference approximations of the time derivative is not directly possible, as the degrees of freedom may get active or inactive across s…
We introduce a symmetric fractional-order reduction (SFOR) method to construct numerical algorithms on general nonuniform temporal meshes for semilinear fractional diffusion-wave equations. By using the novel order reduction method, the governing problem is transformed to an equivalent coupled system, where the explicit orders of time-fractional derivatives involved are all $α/2$ $(1<α<2)$. The linearized L1 scheme and Alikhanov scheme are then proposed on general time meshes. Under some reasonable regularity assumptions and weak restrictions on meshes, the optimal convergence is derived for t…
We study a wave equation with a nonlocal time fractional damping term that models the effects of acoustic attenuation characterized by a frequency dependence power law. First we prove existence of a unique solution to this equation with particular attention paid to the handling of the fractional derivative. Then we derive an explicit time stepping scheme based on the finite element method in space and a combination of convolution quadrature and second order central differences in time. We conduct a full error analysis of the mixed time discretization and in turn the fully space time discretize…
An ADI Crank-Nicolson Orthogonal Spline Collocation Method for the Two-Dimensional Fractional Diffusion-Wave Equation
Mathematical Reviews, American Mathematical Society（美国数学学会数学评论）；College of Mathematics and Computer Science, Hunan Normal University（湖南师范大学数学与计算机科学学院）；School of Mathematics, Central South University（中南大学数学学院）
A Fourier Pseudospectral Method for the "Good" Boussinesq Equation with Second Order Temporal Accuracy
School of Science, Southwest University of Science and Technology（西南科技大学理学院）；Department of Mathematics, University of Tennessee（田纳西大学数学系）；Mathematics Department, University of Massachusetts（马萨诸塞大学数学系）
