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A class of explicit exponential general linear methods. (English) Zbl 1103.65061
The paper presents a class of explicit exponential integrators for semilinear problems $$y'(t) = Ly(t) + N(t,y(t))$$, where $$L$$ is a sectorial linear operator and $$N$$ a smooth nonlinear map. This abstract framework includes semilinear parabolic initial-boundary value problems. The explicit exponential Runge-Kutta and exponential Adams-Bashforth methods are included as special cases in the presented class. This class, moreover, allows for methods of arbitrary high order with good stability properties.
The authors infer the order conditions and their main result proves that the convergence order of the proposed method is essentially minimum of $$P$$ and $$Q+1$$, where $$P$$ and $$Q$$ stand for the quadrature order and the stage order of the method, respectively. A fixed time step is considered throughout the paper except for a short section which is devoted to the generalization to variable stepsize. The theoretically predicted convergence orders are verified by numerical examples for several methods with quadrature orders up to 4 and stage orders up to 3.

##### MSC:
 65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx) 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L05 Numerical methods for initial value problems 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 34G20 Nonlinear differential equations in abstract spaces 35K55 Nonlinear parabolic equations 65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
Expint
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