A new methodology for evaluating unknown parameters in a numerical method for solving a partialdifferential equation is developed. The main result is the construction of several adaptive algorithmsover the global matrix spectrum to yield “optimal” solutions over a parameter spaces. This methodallows that the global matrix becomes in quasi self-adjoin matrix. Many algorithms based on thealgebraic methods, such that Bisection Methods, Newton methods, or Truncated Newton methodshave been explored with successful results. These procedures do not require exact or approximate,analytical solutions of the continuous problem, and are derived from the numerical solution of thealgebraic equation that characterizes the zeros of the imaginary part of the global matrix spectrum.We call the attention that all operations are done over a set of finite-dimensional spaces. Also, a localprocedure has been implemented to multiple spatial dimensions cases. As a simple illustration, it isapplied to the linear advection-diffusion problem to yield an adaptive algebraic method. Thenumerical performance of this method in one and two dimensions is analyzed, and it is found that ityields results that are commensurate to the SUPG method, inclusive in cases of high anisotropy.In the numerical experiments, we have found that the resulting diffusivity, which is a non-linearfunction of the numerical solution, converges to the value obtained using the SUPG analysis. Weshow the convergence of Quasi-Newton method as a function of iteration number, for differentvalues of Peclet numbers and error levels. For an error level equal to 1×10~(-2) in L~2-norm, thisprocedure converges to the SUPG method within about 10 iterations. For an error level equal to1×10~(-8) in L~2-norm, we found that the iteration numbers decrease to 30 iterations while the values ofPeclet numbers increase until 1×10~(10). Applying a Hybrid Method the numerical results wereimproved. We have found successful results for error level lower to 1×10~(-4), combining the BisectionMethod and the Quasi-Newton Method. Also, a method to reduce the phase of largest eigenvalue ofthe resulting global matrix has been examined. Several interesting aspects of the proposedmethodology will be studied in the future. These include a parallel version of this method and the useof gradient methods.
展开▼
