High-Accuracy Spectral-like Legendre–Darboux Method for Initial Value Problems

A high-order single-step implicit method, the Legendre–Darboux Method of order six (LDM6), is introduced for solving both linear and nonlinear initial value problems. Unlike classical Taylor expansions, LDM6 systematically constructs higher-order derivatives via the Darboux formula with Legendre polynomials, yielding a compact scheme of exceptional accuracy and strong stability. To the best of current knowledge, LDM6 is the only single-step method exhibiting spectral-like behavior, achieving near machine-precision global accuracy while retaining efficiency for large step sizes. Comparative experiments on nonlinear cooling problems and the logistic growth model demonstrate that LDM6 surpasses the classical eighth-stage Runge–Kutta method (RK6) in accuracy, stability, and robustness. It attains unprecedented global errors as low as 10−38 and maintains stability for large steps (e.g., h=10), whereas RK6 suffers significant error accumulation. These results establish LDM6 as a uniquely efficient, high-fidelity integrator and the first single-step method with spectral-like accuracy, offering a new paradigm for high-precision time integration.