In this workwe propose and analyze a fully discretemodifiedCrank–Nicolson finite element (CNFE)method with quadrature for solving semilinear second-order hyperbolic initial-boundary value problems. We prove optimal-order convergence in both time and space for the quadrature-modified CNFE scheme that does not require nonlinear algebraic solvers. Finally, we demonstrate numerically the order of c...

In this work, we propose a Crank-Nicolson-type scheme with variable steps for the time fractional Allen-Cahn equation. The proposed is shown to be unconditionally stable (in variational energy sense), and maximum bound preserving. Interestingly, discrete stability result obtained in paper can recover classical dissipation law when order $\alpha \rightarrow 1.$ That is, our asymptotically preser...

We interpret the Cayley transform of linear ( niteor in nitedimensional) state space systems as a numerical integration scheme of Crank Nicolson type. The scheme is known as Tustin's method in the engineering literature, and it has the following important Hamiltonian integrator property: if Tustin's method is applied to a conservative (continuous time) linear system, then the resulting (discret...

We interpret the Cayley transform of linear (finiteor infinite-dimensional) state space systems as a numerical integration scheme of Crank–Nicolson type. The scheme is known as Tustin’s method in the engineering literature, and it has the following important Hamiltonian integrator property: if Tustin’s method is applied to a conservative (continuous time) linear system, then the resulting (disc...

We present an algorithm for the time integration of nonlinear partial differential equations. The algorithm uses distributed approximating functionals, which are based on an analytic approximation method, in order to achieve highly accurate spatial derivatives. The time integration is based on a second-order unconditionally A -stable Crank-Nicolson scheme with a Newton solver. We apply the inte...

<p style='text-indent:20px;'>We consider a hyperbolic free boundary problem by means of minimizing time discretized functionals Crank-Nicolson type. The feature this functional is that it enjoys energy conservation in the absence boundaries, which an essential property for numerical calculations. existence and regularity minimizers shown estimate derived. These results are then used to sh...

We study a generalized extrapolated Crank–Nicolson scheme for the time discretization of a semilinear integro-differential equation with a weakly singular kernel, in combination with a space discretization by linear finite elements. The scheme uses variable grids in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assumin...

In this paper, we study a numerical simulation method for the initial-boundary problem of dissipative of the generalized symmetric regularized long-wave equations with damping term. A nonlinear-implicit Crank-Nicolson finite difference scheme is constructed. The scheme is a twoleveled scheme. The error estimations, convergence, stablity and uniqueness of the solution are proven by discrete ener...