Stability of Runge-Kutta methods for abstract time-dependent parabolic problems: The Hölder case

Abstract

We consider an abstract time-dependent, linear parabolic problem \[ u ′ ( t ) = A ( t ) u ( t ) , u ( t 0 ) = u 0 , u’(t) = A(t)u(t), \qquad u(t_0) = u_0, \] where A ( t ) : D ⊂ X → X A(t) : D \subset X \to X , t ∈ J t \in J , is a family of sectorial operators in a Banach space X X with time-independent domain D D . This problem is discretized in time by means of an A( θ \theta ) strongly stable Runge-Kutta method, 0 > θ > π / 2 0 > \theta >\pi /2 . We prove that the resulting discretization is stable, under the assumption \[ ‖ ( A ( t ) − A ( s ) ) x ‖ ≤ L | t − s | α ( ‖ x ‖ + ‖ A ( s ) x ‖ ) , x ∈ D , t , s ∈ J , \| (A(t) - A(s) )x \| \le L|t-s|^\alpha (\|x\|+ \| A(s)x\|), \qquad x\in D, \,t,\,s \in J, \] where L > 0 L>0 and α ∈ ( 0 , 1 ) \alpha \in (0,1) . Our results are applicable to the analysis of parabolic problems in the L p L^p , p ≠ 2 p \ne 2 , norms.