Hybrid Algorithm for Singularly Perturbed Delay Parabolic Partial Differential Equations

This study aims at constructing a numerical scheme for solving singularly perturbed parabolic delay differential equations. Taylor’s series expansion is applied to approximate the shift term. The obtained result is approximated by using the implicit Euler method in the temporal discretization on a uniform step size with the hybrid numerical scheme consisting of the midpoint upwind method in the outer layer region and the cubic spline method in the inner layer region on a piecewise uniform Shishkin mesh in the spatial discretization. The constructed scheme is an ε−uniformly convergent accuracy of order one. Some test examples are considered to testify the theoretical investigations

Notes on Accuracy of Finite-Volume Discretization Schemes on Irregular Grids

Truncation-error analysis is a reliable tool in predicting convergence rates of discretization errors on regular smooth grids. However, it is often misleading in application to finite-volume discretization schemes on irregular (e.g., unstructured) grids. Convergence of truncation errors severely degrades on general irregular grids; a design-order convergence can be achieved only on grids with a certain degree of geometric regularity. Such degradation of truncation-error convergence does not necessarily imply a lower-order convergence of discretization errors. In these notes, irregular-grid computations demonstrate that the design-order discretization-error convergence can be achieved even when truncation errors exhibit a lower-order convergence or, in some cases, do not converge at all

Some Aspects of Essentially Nonoscillatory (ENO) Formulations for the Euler Equations, Part 3

An essentially nonoscillatory (ENO) formulation is described for hyperbolic systems of conservation laws. ENO approaches are based on smart interpolation to avoid spurious numerical oscillations. ENO schemes are a superset of Total Variation Diminishing (TVD) schemes. In the recent past, TVD formulations were used to construct shock capturing finite difference methods. At extremum points of the solution, TVD schemes automatically reduce to being first-order accurate discretizations locally, while away from extrema they can be constructed to be of higher order accuracy. The new framework helps construct essentially non-oscillatory finite difference methods without recourse to local reductions of accuracy to first order. Thus arbitrarily high orders of accuracy can be obtained. The basic general ideas of the new approach can be specialized in several ways and one specific implementation is described based on: (1) the integral form of the conservation laws; (2) reconstruction based on the primitive functions; (3) extension to multiple dimensions in a tensor product fashion; and (4) Runge-Kutta time integration. The resulting method is fourth-order accurate in time and space and is applicable to uniform Cartesian grids. The construction of such schemes for scalar equations and systems in one and two space dimensions is described along with several examples which illustrate interesting aspects of the new approach

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology

A robust high-resolution hydrodynamic numerical model for surface water flow and transport processes within a flexible software framework

Paralleltitel: Ein robustes hochauflösendes hydrodynamisch-numerisches Modell für Oberflächenabfluss- und Transportprozesse innerhalb eines flexiblen Software-Framework

Absorbing Boundary Conditions and Numerical Methods for the Linearized Water Wave Equation in 1 and 2 Dimensions.

The linearized water wave equation (WWE) models incompressible, irrotational, inviscid free surface flows in deep water. We will investigate the WWE in both one and two spatial dimensions and derive nonreflecting boundary conditions for both. We will calculate numerical solutions for a fractional PDE arising as a nonreflecting boundary condition to the 1-D and 2-D WWE and discuss convergence and stability of the numerical methods. The nonreflecting boundary conditions will be implemented in a boundary layer around the computational domain.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133175/1/dkprigge_1.pd

An adaptively-refined quadtree grid method for incompressible flows

A quadtree grid method used in conjunction with a pressure-based flow solution algorithm for the incompressible Navier-Stokes equations is presented. Two different flow solution methods are studied, each a cell-centered, primitive variable, finite volume procedure based on the SIMPLE algorithm. Solution adaptive grid refinement is used to resolve high-gradient flow regions;The quadtree grid, which is composed of quadrilateral cells that can be subdivided into four quadrants, is examined, and the quadtree data structure, and its advantages when used in the numerical solution of the Navier-Stokes equations, is discussed;Two flow solution methods, including their theoretical formulations, solution procedures, and results obtained for several test cases, are presented. The first solution method approximates the flux across the face between two cells by using flow variable values at points perpendicular to the face. The values at these points are determined through a linear reconstruction from cell-centered values, which results in additions to the source terms of the governing equations. The second flow solution method employs a second-order upwind approximation for the flux across a cell face;The results of the test cases show that an adaptively-refined quadtree grid can yield a better grid distribution over the flow, and therefore give more accurate solutions, as well as improved convergence rates, than can a structured grid with a similar number of grid points