We propose implicit integrators for solving stiff differential equations on unit spheres. Our approach extends the standard backward Euler and Crank-Nicolson methods in Cartesian space by incorporating the geometric constraint inherent to the unit sphere without additional projection steps to enforce the unit length constraint on the solution. We construct these algorithms using the exponential map and spherical linear interpolation (SLERP) formula on the unit sphere. Specifically, we introduce a spherical backward Euler method, a projected backward Euler method, and a second-order symplectic spherical Crank-Nicolson method. While all methods require solving a system of nonlinear equations to advance the solution to the next time step, these nonlinear systems can be efficiently solved using Newton's iterations. We will present several numerical examples to demonstrate the effectiveness and convergence of these numerical schemes. These examples will illustrate the advantages of our proposed methods in accurately capturing the dynamics of stiff systems on unit spheres.
A linearized numerical scheme is proposed to solve the nonlinear time fractional parabolic problems with time delay. The scheme is based on the standard Galerkin finite element method in the spatial direction, the fractional Crank-Nicolson method and extrapolation methods in the temporal direction. A novel discrete fractional Grönwall inequality is established. Thanks to the inequality, the error estimate of fully discrete scheme is obtained. Several numerical examples are provided to verify the effectiveness of the fully discrete numerical method.
We consider a hyperbolic free boundary problem by means of minimizing time discretized functionals of Crank-Nicolson type. The feature of this functional is that it enjoys energy conservation in the absence of free boundaries, which is an essential property for numerical calculations. The existence and regularity of minimizers is shown and an energy estimate is derived. These results are then used to show the existence of a weak solution to the free boundary problem in the 1-dimensional setting.
We provide optimal order pressure error estimates for the Crank-Nicolson semidiscretization of the incompressible Navier-Stokes equations. Second order estimates for the velocity error are long known, we prove that the pressure error is of the same order if considered at interval midpoints, confirming previous numerical evidence. For simplicity we first give a proof under high regularity assumptions that include nonlocal compatibility conditions for the initial data, then use smoothing techniques for a proof under reduced assumptions based on standard local conditions only.
This paper presents a Crank-Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. The spatial discretization adopts the Galerkin finite element method (FEM), and the temporal discretization employs the CNLF method for linear terms and the semi-implicit method for nonlinear terms. The first step uses Stokes style's scheme, the second step employs the Crank-Nicolson extrapolation scheme, and others apply the CNLF scheme. We testify that the fully discrete scheme is stable and convergent when the time step is less than or equal to a positive consta…
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…
In this paper we consider a mass- and energy--conserving Crank-Nicolson time discretization for a general class of nonlinear Schrödinger equations. This scheme, which enjoys popularity in the physics community due to its conservation properties, was already subject to several analytical and numerical studies. However, a proof of optimal $L^{\infty}(H^1)$-error estimates is still open, both in the semi-discrete Hilbert space setting, as well as in fully-discrete finite element settings. This paper aims at closing this gap in the literature. We also suggest a fixed point iteration to solve the a…
Error estimates of the Crank-Nicolson Galerkin method for the time-dependent Maxwell-Schrödinger equations under the Lorentz gauge
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）
In this paper, we develop a linearized fractional Crank-Nicolson-Galerkin FEM for Kirchhoff type quasilinear time-fractional integro-differential equation $\left(\mathcal{D}^α\right)$. In general, the solutions to the time-fractional problems exhibit a weak singularity at time $t=0$. This singular behavior of the solutions is taken into account while deriving the convergence estimates of the developed numerical scheme. We prove that the proposed numerical scheme has an accuracy rate of $O(M^{-1}+N^{-2})$ in $L^{\infty}(0,T;L^{2}(Ω))$ as well as in $L^{\infty}(0,T;H^{1}_{0}(Ω))$, where $M$ and …
A Crank-Nicolson Finite Element Method and the Optimal Error Estimates for the modified Time-dependent Maxwell-Schrödinger Equations
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）；LSEC, NCMIS, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）
We propose and analyze a linearly stabilized semi-implicit diffusive Crank--Nicolson scheme for the Cahn--Hilliard gradient flow. In this scheme, the nonlinear bulk force is treated explicitly with two second-order stabilization terms. This treatment leads to linear elliptic system with constant coefficients and provable discrete energy dissipation. Rigorous error analysis is carried out for the fully discrete scheme. When the time step-size and the space step-size are small enough, second order accuracy in time is obtained with a prefactor controlled by some lower degree polynomial of $1/\var…
We show that the Strang splitting method applied to a diffusion-reaction equation with inhomogeneous general oblique boundary conditions is of order two when the diffusion equation is solved with the Crank-Nicolson method, while order reduction occurs in general if using other Runge-Kutta schemes or even the exact flow itself for the diffusion part. We prove these results when the source term only depends on the space variable, an assumption which makes the splitting scheme equivalent to the Crank-Nicolson method itself applied to the whole problem. Numerical experiments suggest that the secon…
This paper considers a two-step fourth-order modified explicit Euler/Crank-Nicolson numerical method for solving the time-variable fractional mobile-immobile advection-dispersion model subjects to suitable initial and boundary conditions. Both stability and error estimates of the new approach are deeply analyzed in the $L^{\infty}(0,T;L^{2})$-norm. The theoretical studies show that the proposed technique is unconditionally stable with convergence of order $O(k+h^{4})$, where $h$ and $k$ are space step and time step, respectively. This result indicate that the two-step fourth-order formulation …
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）
This paper proposes a replica exchange preconditioned Langevin diffusion discretized by the Crank-Nicolson scheme (repCNLD) to handle high-dimensional and multi-modal distribution problems. Sampling from high-dimensional and multi-modal distributions is a challenging question. The performance of many standard MCMC chains deteriorates as the dimension of parameters increases, and many MCMC algorithms cannot capture all modes if the energy function is not convex. The proposed repCNLD can accelerate the convergence of the single-chain pCNLD, and can capture all modes of the multi-modal distributi…
In this paper we study numerical positivity and contractivity in the infinite norm of Crank-Nicolson method when it is applied to the diffusion equation with homogeneous Dirichlet boundary conditions. For this purpose, the amplification matrices are written in terms of three kinds of Chebyshev-like polynomials, and necessary and sufficient bounds to preserve the desired qualitative properties are obtained. For each spatial mesh, we provide the equations that must be solved as well as the intervals that contain these bounds; consequently, they can be easily obtained by a bisection process. Besi…
This paper develops a robust three-level time split high-order Leapfrog/Crank-Nicolson technique for solving the two-dimensional unsteady sobolev and regularized long wave equations arising in fluid mechanics. A deep analysis of the stability and error estimates of the proposed approach is considered using the $L^{\infty}(0,T;H^{2})$-norm. Under a suitable time step requirement, the theoretical studies indicate that the constructed numerical scheme is strongly stable (in the sense of $L^{\infty}(0,T;H^{2})$-norm), temporal second-order accurate and convergence of order $O(h^{\frac{8}{3}})$ in …
On Error Estimates of the Crank-Nicolson-Polylinear Finite Element Method with the Discrete TBC for the Generalized Schrödinger Equation in an Unbounded Parallelepiped
National Research University Higher School of Economics（国立高等经济大学）
Analysis of fully discrete Crank-Nicolson finite element methods for a stochastic Keller-Segel chemotaxis system with gradient-type multiplicative noise
A fully decoupled and structure-preserving relaxation Crank--Nicolson finite element method for Gross--Pitaevskii--Poisson model
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Hunan International Scientific and Technological Innovation Cooperation Base of Computational Science, Xiangtan University（湘潭大学）；College of Mathematics and Statistics, Hengyang Normal University（衡阳师范学院）；Hunan Research Center of the Basic Discipline Fundamental Algorithmic Theory and Novel Computational Methods, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University（湘潭大学）；Department of Mathematical Sciences, The University of Texas at El Paso（得克萨斯大学埃尔帕索分校）
In this paper, we propose and analyze a fully discrete finite element projection method for the magnetohydrodynamic (MHD) equations. A modified Crank--Nicolson method and the Galerkin finite element method are used to discretize the model in time and space, respectively, and appropriate semi-implicit treatments are applied to the fluid convection term and two coupling terms. These semi-implicit approximations result in a linear system with variable coefficients for which the unique solvability can be proved theoretically. In addition, we use a second-order decoupling projection method of the V…
This work is devoted to an inverse problem of identifying a source term depending on both spatial and time variables in a parabolic equation from single Cauchy data on a part of the boundary. A Crank-Nicolson Galerkin method is applied to the least squares functional with an quadratic stabilizing penalty term. The convergence of finite dimensional regularized approximations to the sought source as measurement noise levels and mesh sizes approach to zero with an appropriate regularization parameter is proved. Moreover, under a suitable source condition, an error bound and corresponding converge…
