Using a variational time discretization of geodesics efficient numerical implementations can be derived.
Using a convenient time discretization of the motion equation of the free boundary, the study of the problem is reduced to treat a stationary one, which is presented as a shape design problem.
We consider the efficient solution of the Cahn Hilliard variational inequality using an implicit time discretization, which is formulated as an optimal control problem with pointwise constraints on the control.
The numerical solution of the aforementioned initial boundary value problem is performed using the analog equation method, a BEM based method, leading to a system of nonlinear differential-algebraic equations (DAE), which is solved using an efficient time discretization scheme.
On the other hand, the immersed boundary method is also known to suffer from a severe timestep stability restriction when using an explicit time discretization.
Application of the boundary element technique yields a system of nonlinear Differential Algebraic Equations, which is solved using an efficient time discretization scheme, from which the transverse and axial displacements are computed.
The resulting differential equation is solved through finite volume discretization using an implicit time discretization scheme.
We then introduce our explicit approach which is based on three main ingredients: first recast the RD discretization as a stabilized Galerkin scheme, then use a shifted time discretization in the stabilization operator, and lastly apply high order mass lumping on the Galerkin component of the discretization.
For a time discretization, Crank Nicolson method is used.
Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate O k3−α), 0<α<1 for any fixed tn>0 for smooth and nonsmooth data in both homogeneous and inhomogeneous cases.
