RBF based backward differentiation methods for stiff differential equations

Numerical solutions of initial value problems (IVPs) for stiff differential equations via explicit methods such as Euler’s method, trapezoidal method and Runge–Kutta methods suffer from stability issues and demand unacceptably small time steps. Backward differentiation formulas (BDF), a class of implicit methods, have been successfully used for resolving stiff IVPs. Classical BDF methods are derived using polynomial basis functions. In this paper, we develop radial basis function based finite difference (RBF-FD) type BDF methods for solving stiff problems. Therefore, we obtain analytical expressions for Gaussian and Multiquadric based RBF-BDF schemes along with their local truncation errors. Then we discuss the stability, order, consistency and convergence of RBF-BDF methods, which also depend on the free shape parameter. Finally, we validate the proposed methods by solving some benchmark problems. In order to gain enhanced accuracy, we adaptively choose the shape parameter such that local truncation error is minimized at each time-step. RBF-BDF methods of order two to six achieve at least one order greater accuracy and order of convergence than corresponding classical BDF schemes.