A novel time integration scheme for linear parabolic PDEs

This paper presents a class of Crank-Nicolson (CN) type schemes enhanced by radial basis function (RBF) interpolation for the time integration of linear parabolic partial differential equations (PDEs). The resulting RBF-CN schemes preserve the structural simplicity of the classical CN method while enabling higher-order temporal accuracy through the optimization of the shape parameter. Consistency analysis shows that the schemes are always at least second-order accurate and a simple choice of the shape parameter increases the accuracy by two orders over the standard CN scheme. Further optimization can reduce the next leading error term to attain even higher-order accuracy. A von Neumann stability analysis confirms that the stability conditions are essentially the same as those of the standard CN scheme. Several numerical experiments in 1D are carried out to verify the theoretical results under different boundary conditions. Particular focus is given to the startup stage, where several strategies for computing the initial steps are examined, and Gauss-Legendre implicit Runge-Kutta (IRK) methods are found to be the most effective. The experiments further demonstrate that, with optimal initialization, the proposed schemes deliver accuracy comparable to implicit Runge-Kutta methods while achieving nearly two orders of magnitude reduction in computational cost.