A variable-θ method for parabolic problems of nonsmooth data

Parabolic initial–boundary value problems with nonsmooth data show either rapid transitions or reduced smoothness in its solution. For those problems, specific numerical methods are required to avoid spurious oscillations as well as unrealistic smoothing of steep changes in the numerical solution. This article investigates characteristics of the θ-method and introduces a variable-θ method as a synergistic combination of the Crank–Nicolson (CN) method (θ=1∕2) and the implicit method (θ=1). It suppresses spurious oscillations, by evolving the solution implicitly at points where the solution shows a certain portent of oscillations or reduced smoothness, and maintains as a similar accuracy as the CN method with smooth data. An effective strategy is suggested for the detection of points where the solution may introduce spurious oscillations (the wobble set); the resulting variable-θ method is analyzed for its accuracy and stability. Various numerical examples are given to verify its effectiveness.