Time-integration scheme Euler implicit

Schlick and Olson recently developed such an algorithm that permits larger time steps. The implicit Euler integration scheme is combined with the Langevin dynamics formulation, which contains frictional and random... 

The volume fraction variables were assumed to be time dependent in the semi-implicit discretization scheme used to solve the continuity equations. Considering the last two terms in the above relation, the transient term was then approximated by an explicit Euler time discretization scheme followed by volume integration in which the transient term was kept constant. If we thereafter multiply the resulting relation by Atjp, the relation can be rewritten as ... 

For the spatial solution of the nonlinear coupled multi-field problem given in Sect 1.3, the Finite Element Method (FEM) is applied. The equations for the three fields are solved with a Newton-Raphson algorithm, and the time integration is performed with the implicit Euler backwards scheme. 

To discretize the continuity, the differential equation is first integrated over the scalar grid cell volume. This is the same grid cell volume as was used for the generalized scalar quantity in Sect. 12.8. By use of the first order implicit Euler time discretization scheme and the basic theorem of integration for the convective fluxes, a discrete form of (12.195) can be expressed as ... 

The continuity equation is re-written on the integral form, integrated in time and over a grid cell volume in the non-staggered grid for the scalar variables sketched in Fig C.2. The transient terms are discretized with the implicit Euler scheme. 

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