Kernel optimum nearly-analytical discretization (KOND) algorithm applied to parabolic and hyperbolic equations

Abstract

Two applications of the Kernel Optimum Nearly-analytical Discretization (KOND) algorithm to the parabolic and the hyperbolic type equations are presented in detail to lead to novel numerical schemes with very high numerical accuracy. It is demonstrated numerically that the two-dimensional KOND-P scheme for the parabolic type yields much less numerical error by over 2–3 orders, measured quantitatively by the root mean square deviation from analytical solutions, and reduces the CPU time to about 1/5 for a common numerical accuracy, compared with the conventional explicit scheme of reference. It is also demonstrated numerically that the KOND-H scheme for the hyperbolic type yields much less diffusive error and has fairly high stability for both of the linear and the nonlinear wave propagations compared with other conventional schemes. Origins of numerical errors in data processing in general numerical schemes are discussed.