On the other hand, using the explicit Euler scheme for the temporal update of the phase-field value, the temporal step (triangle t) depends on the coefficient B and can decrease significantly.
In order to demonstrate the effectiveness of the present shell-based SPH method, several numerical applications involving geometrically nonlinear behavior are carried out using the explicit dynamics scheme for the time integration.
3-D transonic gas flow through the stator and rotor blades in relative motion with periodicity on the whole annulus is described by the unsteady Euler conservation equations, which are integrated using the explicit monotonous finite volume difference scheme of Godunov Kolgan.
Unsteady computations are performed using the explicit (Euler) method, with a streamline upwind scheme for advection term.
However, in their work, they used the explicit numerical scheme for approximated problem which is strongly dependent to the parameter p. To overcome this, we use in this work an efficient algorithm which is a combination of the classical additive operator splitting and a nonlinear relaxation algorithm.
The basic idea of the algorithm is that the time stepping scheme can differ in the blocks of the grid for a given time step: an explicit scheme is used in the blocks where the local stability requirement is not violated and an implicit scheme is used in the blocks where the explicit scheme would be unstable.
For the quanto option pricing model (2D B-S equation), we used the classical implicit scheme, the explicit scheme, and the Crank-Nicolson scheme, constructed a parallel difference scheme-alternating band Crank-Nicolson (ABdC-N) scheme, which is unconditionally stable, and which is close to second-order accuracy.
The explicit scheme was used in the algorithm of numerical calculation in time domain.
Instead of the explicit scheme, which is commonly used in conventional LSM, a semi-implicit additive operator splitting scheme was carried out in our study to solve the LS equation.
