To study the heat or diffusion equation it is often used the Crank-Nicolson method which is unconditionally stable and has order of convergence O(k + h ), where k and h are mesh con2 2 stants. Unfortunately, using this method in conventional floatingpoint arithmetic we get solutions including not only the method error, but also representation and rounding error, Therefore, we propose an interva...

We propose and analyze a linear stabilization of the Crank-Nicolson Leap-Frog (CNLF) method that removes all timestep / CFL conditions for stability and controls the unstable mode. It also increases the SPD part of the linear system to be solved at each time step. We give a proof of unconditional stability of the method as well as a proof of unconditional, asymptotic stability of both the stabl...

The objective of this work was to increase our understanding of how motor patterns are produced during movement tasks by quantifying adaptations in muscle coordination in response to altered task mechanics. We used pedaling as our movement paradigm because it is a constrained cyclical movement that allows for a controlled investigation of test conditions such as movement speed and effort. Alter...

This paper introduces a way of reducing the torque variation in scroll compressors. In a scroll compressor where radial compliance device such as the slider bush is used for the flank sealing, the orbiting scroll member is given the freedom of movement in the radial direction. When such an orbiting scroll member orbits around the nonorbiting scroll member whose center is offset from that of the...

We analyze a single step method for solving second-order parabolic initial–boundary value problems. The method uses a step-doubling extrapolation scheme in time based on backward Euler and a Galerkin approximation in space. The technique is shown to be a second-order correct approximation in time. Since step-doubling can be used as a mechanism for step-size control, the analysis is done for var...

We derive residual-based a posteriori error estimates of optimal order for fully discrete approximations for linear parabolic problems. The time discretization uses the Crank– Nicolson method, and the space discretization uses finite element spaces that are allowed to change in time. The main tool in our analysis is the comparison with an appropriate reconstruction of the discrete solution, whi...

In this paper, we present a new pressure-correction projection scheme for solving the time-dependent Navier-Stokes equations, which is based on the Crank-Nicolson extrapolation method in the time discretization. Error estimates for the velocity and the pressure of semidiscretized scheme are derived by interpreting the projection scheme as second-order time discretization of a perturbed system w...

In this work we generate the numerical solutions of the Burgers’ equation by applying the Crank-Nicolson method directly to the Burgers’ equation, i.e., we do not use Hopf-Cole transformation to reduce Burgers’ equation into the linear heat equation. Absolute error of the present method is compared to the absolute error of the two existing methods for two test problems. The method is also analy...