A hybrid approximation scheme for 1-D singularly perturbed parabolic convection-diffusion problems

Our study is concerned with a hybrid spectral collocation approach to solving singularly perturbed 1-D parabolic convection-diffusion problems. In this approach, discretization in time is carried out with the help of Taylor series expansions before the spectral based on novel special polynomials is applied to the spatial operator in the time step. A detailed error analysis of the presented technique is conducted with regard to the space variable. The advantages of this attempt are presented through comparison of our results in the model problems obtained by this technique and other existing schemes

A hybrid approximation scheme for 1-D singularly perturbed parabolic convection-diffusion problems

Our study is concerned with a hybrid spectral collocation approach to solving singularly perturbed 1-D parabolic convection-diffusion problems. In this approach, discretization in time is carried out with the help of Taylor series expansions before the spectral based on novel special polynomials is applied to the spatial operator in the time step. A detailed error analysis of the presented technique is conducted with regard to the space variable. The advantages of this attempt are presented through comparison of our results in the model problems obtained by this technique and other existing schemes

Numerical solutions of one-dimensional parabolic convection-diffusion problems arising in biology by the Laguerre collocation method

In this work, we present a numerical scheme for the approximate solutions of the one-dimensional parabolic convection-diffusion model problems. Diffusion models form a reasonable basis for studying insect and animal dispersal and invasion, which arise from the question of persistence of endangered species, biodiversity, disease dynamics, multi-species competition so on. Convection diffusion problem is also a form of heat and mass transfer in biological models. The presented method is based on the Laguerre collocation method used for these problems of differential equations. In fact, the approximate solution of the problem in the truncated Laguerre series form is obtained by this method. By substituting truncated Laguerre series solution into the problem and by using the matrix operations and the collocation points, the suggested scheme reduces the problem to a linear algebraic equation system. By solving this equation system, the unknown Laguerre coecients can be computed. The accuracy and the efficiency of the method is showed by numerical examples and the comparisons by the other methods

Hermite interpolant multiscaling functions for numerical solution of the convection diffusion equations

A numerical technique based on the Hermite interpolant multiscaling functions is presented for the solution of Convection-diusion equations. The operational matrices of derivative, integration and product are presented for multiscaling functions and are utilized to reduce the solution of linear Convection-diusion equation to the solution of algebraic equations. Because of sparsity of these matrices, this method is computationally very attractive and reduces the CPU time and computer memory. Illustrative examples are included to demonstrate the validity and applicability of the new technique

On Discontinuous Galerkin Methods for Singularly Perturbed and Incompressible Miscible Displacement Problems

This thesis is concerned with the numerical approximation of problems of fluid flow, in particular the stationary advection diffusion reaction equations and the time dependent, coupled equations of incompressible miscible displacement in a porous medium. We begin by introducing the continuous discontinuous Galerkin method for the singularly perturbed advection diffusion reaction problem. This is a method which coincides with the continuous Galerkin method away from internal and boundary layers and with a discontinuous Galerkin method in the vicinity of layers. We prove that this consistent method is stable in the streamline diffusion norm if the convection field flows non-characteristically from the region of the continuous Galerkin to the region of the discontinuous Galerkin method. We then turn our attention to the equations of incompressible miscible displacement for the concentration, pressure and velocity of one fluid in a porous medium being displaced by another. We show a reliable a posteriori error estimator for the time dependent, coupled equations in the case where the solution has sufficient regularity and the velocity is bounded. We remark that these conditions may not be attained in physically realistic geometries. We therefore present an abstract approach to the stationary problem of miscible displacement and investigate an a posteriori error estimator using weighted spaces that relies on lower regularity requirements for the true solution. We then return to the continuous discontinuous Galerkin method. We prove in an abstract setting that standard (continuous) Galerkin finite element approximations are the limit of interior penalty discontinuous Galerkin approximations as the penalty parameter tends to infinity. We then show that by varying the penalization parameter on only a subset of the domain we reach the continuous discontinuous method in the limit. We present numerical experiments illustrating this approach both for equations of non-negative characteristic form (closely related to advection diffusion reaction equations) and to the problem of incompressible miscible displacement. We show that we may practically determine appropriate discontinuous and continuous regions, resulting in a significant reduction of the number of degrees of freedom required to approximate a solution, by using the properties of the discontinuous Galerkin approximation to the advection diffusion reaction equation. We finally present novel code for implementing the continuous discontinuous Galerkin method in C++

Space-time Methods for Time-dependent Partial Differential Equations

Modern discretizations of time-dependent PDEs consider the full problem in the space-time cylinder and aim to overcome limitations of classical approaches such as the method of lines (first discretize in space and then solve the resulting ODE) and the Rothe method (first discretize in time and then solve the PDE). A main advantage of a holistic space-time method is the direct access to space-time adaptivity and to the backward problem (required for the dual problem in optimization or error control). Moreover, this allows for parallel solution strategies simultaneously in time and space. Several space-time concepts where proposed (different conforming and nonconforming space-time finite elements, the parareal method, wavefront relaxation etc.) but this topic has become a rapidly growing field in numerical analysis and scientific computing. In this workshop the focus is the development of adaptive and flexible space-time discretization methods for solving parabolic and hyperbolic space-time partial differential equations