The Krylov subspace approximation techniques described by Gallopoulos and Saad 2] for the numerical solution of parabolic partial diierential equations are extended. By combining the weighted quadrature methods of Lawson and Swayne 6] with Krylov subspace approximations, three major improvements are made. First, problems with time-dependent sources or boundary conditions may be solved more eeciently. Second, methods are derived which have the stability properties (such as A{stability) of the underlying rational approximation to the exponential function. Third, it is possible to present methods which are robust under space discretization reenement. In particular, a xed precision is essentially maintained for the same time integration method and for constant values of the parameters, when the spatial resolution is increased.
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