Introduction to functional analysis, 2nd ed.
Introduction to functional analysis, 2nd ed.
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Some problems of well-posedness of difference schemes on non-uniform grids
Computational Mathematics and Mathematical Physics
On the stability of variable stepsize rational approximations of holomorphic semigroups
Mathematics of Computation
Runge-Kutta approximation of quasi-linear parabolic equations
Mathematics of Computation
Interior estimates for time discretizations of parabolic equations
NUMDIFF-7 Selected papers of the seventh conference on Numerical treatment of differential equations
Applied Numerical Mathematics - Special issue celebrating the centenary of Runge-Kutta methods
Stability of Time-Stepping Methods for Abstract Time-Dependent Parabolic Problems
SIAM Journal on Numerical Analysis
The stability of difference schemes of second-order of accuracy for hyperbolic-parabolic equations
Computers & Mathematics with Applications
Stability of difference schemes for hyperbolic-parabolic equations
Computers & Mathematics with Applications
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We consider variable stepsize Runge-Kutta methods for semilinear evolution equations with a sectorial operator in the linear part. For nonsmooth initial data error estimates are derived that show the interplay of weak singularities and the classical order of convergence. There are no uniformity assumptions on the stepsizes, but we assume a Lipschitz condition for the nonlinearity and a stability function for the method that is less than 1 on the critical sector and vanishes at infinity. Using an extended operational calculus the proof combines a rearrangement trick with a discrete Gronwall estimate including weak singularities. Our main theorem complements respectively extends well-known results of Bakaev, Gonzalez, Lubich, Ostermann and Palencia.