We show a theorem on convergence for finite difference schemes 
applied to the Cauchy problems for the linear Kowalevskian 
systems. 
  We have a nice theorem of Lax that convergence and stability
are equivalent to each other for the finite difference schemes
applied to the linear well-posed Cauchy problems, and we also 
have a clear theorem of Kovalevskaya that there exists a unique
solution in the class of analytic functions for the Cauchy 
problems, whether well-posed or ill-posed, for the Kowalevskian 
systems. We have had no theories in finite difference methods 
for ill-posed Cauchy problems, and we give an affirmative result
to them.
  We reduce the problems to abstract Cauchy problems in Banach 
scales, and we give a convergence estimate for discretization
by the finite difference method. We understand that convergence 
and stability are independent of each other in general, and
we can show convergence even for ill-posed problems for consistent
schemes. Some numerical solutions on multiple-precision arithmetic
are also shown. 
  Some inverse problems are reduced to ill-posed Cauchy problems,
and the theorem gives a clue to construct reliable numerical 
solutions for them.