Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method

Multiquadric (MQ) collocation method is highly efficient for solving partial differential equations due to its exponential error convergence rate. A special feature of the method is that error can be reduced by increasing the value of shape constant c   in the MQ basis function, without refining the grid. It is believed that in a numerical solution without roundoff error, infinite accuracy can be achieved by letting c→∞c→∞. Using the arbitrary precision computation, this paper tests the above conjecture. A sharper error estimate than previously obtained is presented. A formula for a finite, optimal c value that minimizes the solution error for a given grid size is obtained. Using residual errors, constants in error estimate and optimal c formula can be obtained. These results are supported by numerical examples.