Implicit schemes for solving large-scale linear differential–algebraic systems with constant coefficients necessitate at each integration step the solution of a linear system, typically obtained by a Krylov subspace method such as GMRES. To accelerate the convergence, an approach is proposed that computes good initial guesses for each linear system to be solved in the implicit scheme. This approach requires, at each integration step, a small dimensional subspace where a good initial guess is found using the Petrov–Galerkin process. It is shown that the residual associated with the computed initial guess depends on the dimension of the subspace, the order of the implicit scheme, and the discretization stepsize. Several numerical illustrations are reported.

求解具有常数系数的大规模线性微分-代数系统的隐式方案需要在每个积分步骤中求解一个线性系统，通常通过克雷洛夫子空间方法（如GMRES）获得该系统。为了加快收敛速度​​，提出了一种为隐式方案中要解决的每个线性系统计算良好的初始猜测的方法。这种方法在每个集成步骤都需要一个小尺寸的子空间，使用Petrov-Galerkin过程可以找到一个很好的初始猜测。结果表明，与计算出的初始猜测相关的残差取决于子空间的维数，隐式方案的阶数和离散化步长。报告了几个数字插图。