Block Implicit Methods with L-stability for Parabolic Problems

Block implicit methods (BIM) are a group of time integration schemes with desirable stability properties, high-order of accuracy and the capability to compute solutions across multiple time steps simultaneously without numerous initial values. In this paper, some BIM with L-stability are developed, similar to Runge–Kutta methods, a tableau including two matrices and two vectors defines a particular BIM with the required order of accuracy and stability properties. In particular, a type of L-stable BIM with a positive definite matrix and a positive diagonal matrix is introduced. In this study, we also extend the classical finite element theory, commonly for parabolic problems discretized using the Backward Euler or Crank–Nicolson schemes, to this type of BIM. Additionally, in order to solve the resulting large sparse linear system of equations, several tensor structure preserving domain decomposition preconditioners for Krylov subspace methods are also introduced. Finally, some numerical results are reported to demonstrate the effectiveness of the new methods.