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Abstract

Applications of a high order finite difference method with free quadrature points to Cauchy problems of two-dimensional Laplace equation Kentaro IijimaWe construct a high order finite difference method in which the quadrature points can be set at arbitrary locations.It is known that the Cauchy problem of the Laplace equation is unstable with respect to L2-norm.We apply our method to Cauchy problems of the Laplace equation.The finite difference approximation in this research is formulated as follows: the derivative of a function is replaced with a linear combination of values of the function for each quadrature points which are located arbitrarily.We propose an algorithm in order to determine weights in this linear combination.In numerical experiments we obtain a highly precise numerical solution.These results imply effectiveness on solving the Cauchy problem of the Laplace equation by our method.