A class of explicit exponential general linear methods.

*(English)*Zbl 1103.65061The 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.

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.

Reviewer: Tomas Vejchodsky (Praha)

##### 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 |

##### Keywords:

exponential integrators; general linear methods; explicit schemes; abstract evolution equations; semilinear parabolic problems; convergence; high-order methods; Runge-Kutta method; Adams-Bashforth methods; stability; variable stepsize; numerical examples##### Software:

Expint
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DOI

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