Kansa method

The Kansa method is a computer method used to solve <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equations</a>. Its main advantage is it is very easy to understand and program on a computer. It is much less complicated than the <a href="/facts/Finite_element_method/eUcfP1vn">finite element method</a>. Another advantage is it works well on multi variable problems. The finite element method is complicated when working with more than 3 space variables and time.
The Kansa Method can be explained by an analogy to a basketball court with many light bulbs suspended all across the ceiling. You solve for the brightness of each bulb so that the desired light intensity directly on the floor of the basketball court under each bulb solves the differential equation at that point. So if the basketball court has 100 bulbs suspended over it; the light intensity at any point on the floor of the basketball court
approaches a light intensity that approximately solves the differential equation at any location on the floor of the basketball court. A simple computer program can solve by iteration for the brightness of each bulb, which makes this method easy to program. This method does not need weighted residuals (Galerkin), integration, or advanced mathematics.
E. J. Kansa in very early 1990s made the first attempt to extend <a href="/facts/Radial_basis_function/QnWM1mBN">radial basis function</a> (RBF), which was then quite popular in scattered data processing and function approximation, to the solution of <a href="/facts/Partial_differential_equations/DyPsBlv7">partial differential equations</a> in the strong-form collocation formulation. His RBF collocation approach is inherently meshless, easy-to-program, and mathematically very simple to learn. Before long, this method became known as the Kansa method in the academic community.
Because the RBF uses the one-dimensional Euclidean distance variable irrespective of dimensionality, the Kansa method is independent of dimensionality and geometric complexity of problems of interest. The method is a domain-type numerical technique in the sense that the problem is discretized not only on the boundary to satisfy boundary conditions but also inside domain to satisfy governing equation.
In contrast, there is another type of RBF numerical methods, called boundary-type RBF collocation method, such as the <a href="/facts/Method_of_fundamental_solution/RP4a1cq3">method of fundamental solution</a>, <a href="/facts/Boundary_knot_method/GjAq8xoy">boundary knot method</a>, <a href="/facts/Singular_boundary_method/7tm3TA7H">singular boundary method</a>, <a href="/facts/Boundary_particle_method/csI5W3xM">boundary particle method</a>, and regularized meshless method, in which the basis functions, also known as kernel function, satisfy the governing equation and are often fundamental solution or general solution of governing equation. Consequently, only boundary discretization is required.
Since the RBF in the Kansa method does not necessarily satisfy the governing equation, one has more freedom to choose a RBF. The most popular RBF in the Kansa method is the multiquadric (MQ), which usually shows spectral accuracy if an appropriate shape parameter is chosen.

Kansa method open-in-new

Kansa method