In the numerical experiments, the multisymplectic concatenated midpoint scheme (a centered cell discretization) is shown to preserve the local conservation laws extremely well over long times and to preserve global invariants such as the norm and momentum within roundoff.
To calculate the next status of spring forces, positions, and velocities, we applied and judged three types of numerical integration with semi-Euler, midpoint, and 4th-order Runge-Kutta method based on CPU only, multi-threaded CPU, GPU, and GPU with multi-threaded CPU.
Numerical approximation of modes using the midpoint method is applied to study the modes.
By using the method of characteristics we can estimate the error as follows ‖ u (τ, x ) − u 1 ‖ ≤ τ L ‖ u (τ 2, 0 ) − 1 τ ∫ 0 τ u (t, 0 ) d t ‖, where L is the Lipschitz constant of u 0. Therefore, the numerical scheme can be seen as the midpoint rule approximating an integral.
We simulate some example systems for various stepsizes τ over time t = [0, T] using three fixed-step numerical methods: the Euler τ-leap (ETL), midpoint τ-leap (MPTL) and θ-trapezoidal τ-leap (TTTL) methods (with θ = 0.55), and their extrapolated versions, the xETL, xMPTL and xTTTL.
The proposed method is completed with the midpoint rule for time integration and numerical results are provided, including considerations for interconnection and closed loop behaviors and isotropy comparison between the proposed meshes.
Cao et al. [16] studied the implicit midpoint method and constructed a new numerical scheme for modified fractional sub-diffusion equation with nonlinear source term.
The available numerical integration methods in FORG3D are Euler method, Midpoint method and the fourth-order Runge-Kutta method [ 15].
The implicit midpoint rule is one of the powerful numerical methods for solving ordinary differential equations and differential algebraic equations.
In our numerical investigations we show results for the ETL, the midpoint τ-leap (MPTL) and the θ-trapezoidal τ-leap (TTTL) method.
The implicit midpoint rule (IMR) is one of the powerful numerical methods for solving ordinary differential equations (in particular, the stiff equations) [1 6] and differential-algebra equations [7].
