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Modified Iterated Crank-Nicolson Method with Improved Accuracy

The iterated Crank-Nicolson (ICN) method is a successful numerical algorithm in numerical relativity for solving partial differential equations. The $θ$-ICN method is the extension of the original ICN method where $θ$ is the weight when averaging the predicted and corrected values. It has better stability when $θ$ is chosen to be larger than 0.5, but the accuracy is reduced since the $θ$-ICN method is second order accurate only when $θ$ = 0.5. In this paper, we propose two modified $θ$-ICN algorithms that have second order of convergence rate when $θ$ is not 0.5, based on two different ways to choose the weight $θ$. The first approach employs two geometrically averaged $θ$s in two iterations within one time step, and the second one uses arithmetically averaged $θ$s for two consecutive time steps while $θ$ remains the same in each time step. The stability and second order accuracy of our methods are verified using stability and truncation error analysis and are demonstrated by numerical examples on linear and semi-linear hyperbolic partial differential equations and Burgers' equation.