Idempotent filtering in spectral and spectral element methods

The comparison of numerical results for implicit–explicit and fully explicit Runge–Kutta time integration methods for a nozzle flow problem shows that filtering can significantly degrade the accuracy of the numerical solution for long-time integration problems. We demonstrate analytically and numerically that filtering-in-time errors become additive for ∥uN(x, t + kΔt) − uN(x, t)∥ ≪ ∥uN(x, t)∥ when nonidempotent filters are used, and suggest the development and implementation of idempotent filters.