Spectral Method In Unbounded Domain

Sprayforming is a new technology in metallurgical industry, which have been well applied, and spray atomization is a key technology for spayforming. So the research of spray atomization is very important. But it should be pointed out that we know about the mechanism of spray atomization not enough, since physical phenomenon of spray atomization is very complicated. Because there is large difficulty in theory analysis and experimental study, Use high-accuracy numerical method as means of studying to phenomenon of spray atomization is very necessary. So as a high-accuracy numerical method, spectral method is very important for studying the mechanism of spray atomization.The major difficulty we will meet in studying spray atomization is how to handle boundary condition, such as unbounded domain problem and interface condition. In the thesis, we will major in the method to deal with unbounded domain problem. There are three basic ways to deal with unbounded domain problem, such as domain truncated, expand in functions whose variables are infinite or semi-infinite, and coordinate transformation. We develop a new exponential transform method combined with coordinate transformation to deal with unbounded domain problem. By using of the method, we computed a linear problem, the second kind of modified Bessel Function Kn (z), and a nonlinear problem, Burgers Equation in unbounded domain.The second kind of modified Bessel function Kn(z) is decay exponentially when z - +. So if itis approximated by Chebyshev polynomial, we will meet large error. In the thesis, We develop a new exponential transform method combined with Rational Chebyshev Polynomials or Chebyshev Spectral Collocation Method to compute the second kind of modified Bessel function. By introducing an exponential transformation to achieve high precision and the property of second kind of modified Bessel Function Kn(z) is identified with Chebyshev polynomial approximation, so we can approximate itefficiently.We also use the otponential transformation combined with coordinate transformation to deal with a nonlinear problem, Burgers Equation. First, we deal with Burgers Equation and boundary condition by the exponential transformation; then through algebraic mapping or logarithmic mapping, we map the semi-infinite into finite one; at last, we solve it by spectral method. We found that using algebraic mapping to solve Burgers Equation is not accuracy enough and convergent slowly. But by using logarithmic mapping to solve BurgersEquation, the compute errors are small. Parameter A/ is key to success of the method.The main difference between the method we developed and another ones is that we introduce a new exponential transformation to transform problem with considering the function /( z] is decayexponentially when z - + . And it should be pointed out that the method we develop is not only suit for deal with the linear problem, the second kind of modified Bessel Function Kn(z), and thenonlinear problem, Burgers Equation in unbounded domain, but also supply a new way to solve another problems in unbounded domain.