and Applied Analysis 3 3. Numerical Solution Environmental phenomena, such as groundwater flow described by variational order derivative, are highly complex phenomena, which do not lend themselves readily to analysis of analytical models. The discussion presented in this section will therefore be devoted to the derivation of numerical solution to the modified Hantush equation (6). Solving diffi...

Pressure correction schemes and the method of characteristics are combined to obtain numerical approximation procedures for incompressible Navier-Stokes ows on massive parallel computers. The projection step is carried out with a new more eecient parallel preconditioned conjugate gradient method and for the computation of the characteristics we use a fast local Crank-Nicholson solver for linear...

We present a high resolution time integration method for the spherical harmonics equations based on the L-Trap method of Duraisamy, et al. This method is equivalent to the Crank-Nicolson method in smooth regions of the solution and changes form in non-smooth regions of the solution to avoid artificial oscillations in the solution. The method is nonlinear in the sense that the time integration u...

We present a parallel time-stepping method for fluid-structure interactions. The interaction between the incompressible Navier-Stokes equations and hyperelastic solid is formulated in fully monolithic framework. Discretization space based on equal order finite element all variables variant of Crank-Nicolson scheme used as second time integrator. To accelerate solution systems, we analyze parall...

The unsteady stagnation point flow and heat transfer with prescribed flux towards a stretching and shrinking sheet with viscous dissipation is studied. Similarity transformation is adopted to initially convert the governing differential equations into nonlinear ordinary differential equations. The two-point boundary value ordinary differential equations (ODE) are subsequently converted into par...

We consider numerical methods for solving the fractional-in-space Allen-Cahn (FiSAC) equation which contains small perturbation parameters and strong noninearilty. Standard fully discretized schemes for the the FiSAC equation will be considered, namely, in time the conventional first-order implicit-explicit scheme or second-order Crank-Nicolson scheme and in space a secondorder finite differenc...

We present an unconditionally stable second-order hybrid numerical method for solving the Allen–Cahn equation representing amodel for antiphase domain coarsening in a binary mixture. The proposed method is based on operator splitting techniques. The Allen–Cahn equation was divided into a linear and a nonlinear equation. First, the linear equation was discretized using a Crank–Nicolson scheme an...

Three new fully implicit methods which are based on the (5,5) Crank–Nicolson method, the (5,5) N-H (Noye–Hayman) implicit method and the (9,9) N-H implicit method are developed for solving the heat equation in two dimensional space with non-local boundary conditions. The latter is fourth-order while the others are second-order. While the implicit methods developed here, like the scheme based on...