In the present work, three-step Taylor Galerkin finite element method(3TGFEM) and least-squares finite element method(LSFEM) have been discussed for solving parabolic singularly perturbed problems. For singularly perturbed problems, a small parameter called singular perturbation parameter is multiplied with the highest order derivative term. As this singular perturbation parameter approaches to zero, a very sharp change occurs in the solution, which makes it difficult to find solution by traditional methods unless some special treatment is employed. A comparison on the performance of the three schemes namely, (a) 3TGFEM with exponentially fitted splines, (b) explicit least-squares finite element method with linear basis functions and (c) 3TGFEM with linear basis functions, for solving the parabolic singularly perturbed problems has been made. For all the three schemes Shishkin based logarithmic mesh has been used for numerical computations. It has been found out that the 3TGFEM scheme with exponentially fitted splines provides more accurate results as compared to the other two schemes. Detailed error estimates for the three-step Taylor Galerkin scheme with exponentially fitted splines have been presented. The scheme is shown to be conditionally uniform convergent. It is third order accurate in time variable and linear in space variable. Numerical results have been presented for all the three schemes for both linear and non-linear problems.

A novel numerical flux for the Euler equations which is consistent for kinetic energy and entropy condition was proposed recently [1]. This flux makes use of entropy variable based matrix dissipation which can be shown to satisfy an entropy inequality. For hypersonic flows a blended scheme is proposed which gives carbuncle free solutions for blunt body flows while still giving accurate resolution of boundary layers. Several numerical results on standard test cases using high order accurate reconstruction schemes are presented to show the performance of the new schemes.

We present a new time discretization scheme based on the continuous Galerkin Petrov method of polynomial order 3 (cGP(3)-method) which is combined with a reduced numerical time integration (3-point Gauß-Lobatto formula). The solution of the new approach can be computed from the solution of the lower order cGP(2)-method, which requires to solve a coupled 2 × 2 block system on each time interval, followed by a simple post-processing step, such that we get the higher accuracy of 4th order in time in the standard L²-norm with nearly the cost of the cGP(2)-method. Moreover, the difference of both solutions can be used as an indicator for the approximation error in time. For the approximation in space we use the nonparametric \tilde{Q}_3-element which belongs to a family of recently derived higher order nonconforming finite element spaces and leads to an approximation error in space of order 4, too, in the L²-norm. The expected optimal accuracy of the full discretization error in the L²-norm of 4th order in space and time is confirmed by several numerical tests. We discuss implementation aspects of the time discretization as well as efficient multigrid methods for solving the resulting block systems which lead to convergence rates being almost independent of the mesh size and the time step. In our numerical experiments we compare different higher order spatial and temporal discretization approaches with respect to accuracy and computational cost.

We study a model of continuous sedimentation. Under idealizing assumptions, the settling of the solid particles under the influence of gravity can be described by the initial value problem for a one-dimensional scalar conservation law with a flux function that depends discontinuously on the spatial position. There exists several entropy conditions related to the same conservation law in the literature giving rise to uniqueness. The same initial data may give rise to different entropy solutions, depending on the criteria one selects. This motivated us to derive the PDE together with an entropy condition as a hydrodynamic limit from a microscopic interacting particle system.We are inclined to prefer the entropy solution selected by this method. It turns out that this is an entropy condition suggested by Audusse and Pethame in a different context.

In this work, a local algorithm is proposed to optimize the shape parameter of the infinitely smooth Radial Basis Function (RBF) in the local Hermite-RBF (LHRBF) grid free scheme which has been developed to the solve the partial differential equations, in particular, steady Convection Diffusion Equations (CDE), more accurately. The algorithm is based on the re-construction of the forcing term and its (differential) operator value over the centers in the local support domain. A cost function which has similar variation as RMS error function is obtained and the same is minimized to generate a variable shape parameter for each reference center. The LHRBF scheme with the obtained variable (optimal) shape parameter is tested over one and two dimensional steady CDEs to demonstrate its robustness.

Graphics Processing Units (GPUs), originally developed for computer games, now provide computational power for scientific applications. A study on the comparison of computational speed-up and efficiency of a GPU with a CPU for the Finite Pointset Method (FPM), which is a numerical tool in Computational Fluid Dynamics (CFD) is presented. As FPM is based on the point cloud, it is so expensive when the number of particles are in millions. We have demonstrated the application of the FPM using a single-GPU (Nvidia Tesla M2050) and Intel CPU (Dual Xeon). Importance of the GPU is realized by the FPM since GPU yields a computational speed-up of 70× for the Poisson equation with various boundary conditions.

The aquifer thermal energy storage system (ATES) owing to growing demands for sustainable energy has become a popular technology in last few decades for long term storing of excess thermal energy. The efficiency of such a system depends entirely on the capacity of the aquifer to retain heat and hence modeling the heat transfer and transient temperature distribution in the aquifer due to hot water injection is essential. The present study is concerned about presenting a two-dimensional numerical model for such an ATES system with block heterogeneities where hot water is injected through an injection well into the heterogeneous porous aquifer with lesser initial temperature. The transient temperature distribution in the porous aquifer or the advancement of the hot water thermal front generated due to hot water injection into the porous aquifer is modeled here using the multiphysics numerical code COMSOL. First the model is developed for the general case including the convective and the conductive heat transport. Since modeling of such complex systems are associated with numerical errors which may lead to possible errors in estimation of temperature distribution in the aquifer, the results derived numerically afterwards are compared and validated with an analytical model derived by the authors.