A Modified Fourth-Order Compact Crank--Nicolson Scheme with Reduced Truncation Error for the One-Dimensional Heat Equation

Source abstract

The classical Crank–Nicolson scheme is a reliable workhorse for solving the heat equation, offering both unconditional stability and second-order accuracy. However, its spatial accuracy is limited to second order, which often forces the use of extremely fine meshes in applications requiring high precision. Compact finite difference methods overcome this limitation by achieving fourth-order accuracy on compact stencils. In this paper, we modify the classical compact scheme by introducing a tunable correction coefficient and determining its optimal value through numerical optimisation. To the best of our knowledge, this is the first work that combines an asymptotic motivation for coefficient modification, a systematic numerical optimisation, and a complete theoretical analysis for the standard one-dimensional heat equation. We begin by deriving an asymptotic expression, beta = 1 + h²/(30 alpha) + O(h⁴), using Taylor expansion and a smoothness approximation. A systematic numerical search is then carried out for beta in the interval [0.90, 1.10], with the aim of finding the minimum average L-infinity error for several grid sizes. This leads to a consistent reduction of the leading truncation error by approximately 30% compared to the classical compact scheme, while retaining the tridiagonal structure and computational efficiency. A full mathematical analysis is provided, including the local truncation error, von Neumann stability analysis, and convergence via energy estimates. The predicted convergence rates are verified through a variety of numerical experiments conducted on benchmark problems involving variable coefficients, source terms, and stiff problems, with the codes demonstrating improved accuracy. All results are reproducible, and the accompanying Python implementation is provided.