A-Stable High-Order Block Implicit Methods for Parabolic Equations.

In this paper, we consider the time integration of parabolic equations with block implicit methods (BIM). Depending on the size of the block, high-order BIM with A-stability are designed without the need of multiple initial guesses. Similar to Runge--Kutta methods, a BIM can be defined by a tableau including two matrices and two vectors. In addition to the general methodology of BIM, we show a special scheme defined by a positive definite matrix and a positive diagonal matrix; both matrix properties are desirable but not available in Runge--Kutta methods. Moreover, we show that the traditional finite element theory for parabolic problems discretized by the backward Euler or Crank-Nicolson schemes can also be extended to BIM. Finally, we introduce some domain decomposition preconditioners for the linear systems of algebraic equations arising from the block implicit discretization in time and the finite element in space. Some numerical results are also reported to show the effectiveness of BIM.