Spectral methods for modeling supersonic chemically reacting flow fields

A numerical algorithm was developed for solving the equations describing chemically reacting supersonic flows. The algorithm employs a two-stage Runge-Kutta method for integrating the equations in time and a Chebyshev spectral method for integrating the equations in space. The accuracy and efficiency of the technique were assessed by comparison with an existing implicit finite-difference procedure for modeling chemically reacting flows. The comparison showed that the procedure presented yields equivalent accuracy on much coarser grids as compared to the finite-difference procedure with resultant significant gains in computational efficiency

On the fourth-order accurate compact ADI scheme for solving the unsteady Nonlinear Coupled Burgers' Equations

The two-dimensional unsteady coupled Burgers' equations with moderate to severe gradients, are solved numerically using higher-order accurate finite difference schemes; namely the fourth-order accurate compact ADI scheme, and the fourth-order accurate Du Fort Frankel scheme. The question of numerical stability and convergence are presented. Comparisons are made between the present schemes in terms of accuracy and computational efficiency for solving problems with severe internal and boundary gradients. The present study shows that the fourth-order compact ADI scheme is stable and efficient

Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations

Excerpt: Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics

Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit

Spectral method for the unsteady incompressible Navier-Stokes equations in gauge formulation

A spectral method which uses a gauge method, as opposed to a projection method, to decouple the computation of velocity and pressure in the unsteady incompressible Navier-Stokes equations, is presented. Gauge methods decompose velocity into the sum of an auxilary field and the gradient of a gauge variable, which may, in principle, be assigned arbitrary boundary conditions, thus overcoming the issue of artificial pressure boundary conditions in projection methods. A lid-driven cavity flow is used as a test problem. A subtraction method is used to reduce the pollution effect of singularities at the top corners of the cavity. A Chebyshev spectral collocation method is used to discretize spatially. An exponential time differencing method is used to discretize temporally. Matrix diagonalization procedures are used to compute solutions directly and efficiently. Numerical results for the flow at Reynolds number Re = 1000 are presented, and compared to benchmark results. It is shown that the method, called the spectral gauge method, is straightforward to implement, and yields accurate solutions if Neumann boundary conditions are imposed on the gauge variable, but suffers from reduced convergence rates if Dirichlet boundary conditions are imposed on the gauge variable

A Mathematical Model for Epitaxial Semiconductor Crystal Growth from the Vapor Phase on a Masked Substrate

Certain materials used in lasers are made by a process called epitaxial semiconductor crystal growth. In this report a mathematical model is developed for this growth process which occurs on a substrate at the junction between a masked region and exposed substrate in a vapor. This new model consists of two partial differential equations; one for the surface dynamics and one for the crystal growth on the exposed substrate. An analysis of the steady state solutions is furnished. Approximate solutions for time-dependent cases are found using two numerical methods. An asymptotic analysis is also carried out to determine transient solution behavior. The undesireable "bump" structure at the mask/substrate junction which has been observed experimentally is present in the solutions found by each method

A Rotating-Grid Upwind Fast Sweeping Scheme for a Class of Hamilton-Jacobi Equations

We present a fast sweeping method for a class of Hamilton-Jacobi equations that arise from time-independent problems in optimal control theory. The basic method in two dimensions uses a four point stencil and is extremely simple to implement. We test our basic method against Eikonal equations in different norms, and then suggest a general method for rotating the grid and using additional approximations to the derivatives in different directions in order to more accurately capture characteristic flow. We display the utility of our method by applying it to relevant problems from engineering