# unconditionally stable schemes better than conditionally stable ones in accuracy?

Let's consider two finite difference schemes for PDEs/ODEs. One is conditionally stable, the other is unconditionally stable. People always prefer unconditionally stable ones to conditionally stable ones because for the former one can take large steps. However, one needs to take small steps to get higher accuracy anyway. Is it possible that, for the same continuous equation to solve and under the same step length (in the stable regime for the conditionally stable scheme), a conditionally stable scheme is more accurate than an unconditionally stable one? If so, examples?

• Sure, for a 1D diffusion equation an explicit time integration scheme would have a CFL constraint on the time step, but an implicit scheme would not. But an explicit high-order (in time and space) scheme would be [formally] more accurate (within its range of stability) than an implicit low-order scheme. Jan 25, 2023 at 15:47

This is absolutely possible. For example, say you're solving the advection equation $$\frac{\partial q}{\partial t} + \nabla\cdot q\,\mathbf u = 0$$ for the field $$q$$ using your favorite technique for spatial discretization. One of the 3rd-order strong stability-preserving Runge-Kutta methods is going to be more accurate than, say, a 1st-order implicit scheme despite the fact that you can take longer timesteps with the implicit scheme. On top of that, the fact that you can take longer timesteps with an implicit method doesn't necessarily mean that you'll get a very good answer, it just means that you won't get an array of NaNs.
• Ok, what about the implicit midpoint method $(z_{n + 1} - z_n) / \delta t = f((z_{n + 1} + z_n) / 2)$? What is its stability region? Is it dissipative? Is it symplectic? Jan 26, 2023 at 4:36