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On optimal high-order in time approximations for the Korteweg-de Vries equation. (English) Zbl 0725.65107
This paper discusses the computation of periodic solutions to the Korteweg-de Vries equation \(u_ t+uu_ x+\epsilon u_{xxx}=0,\) \(x\in (0,1)\), \(t\in (0,\infty)\) with initial value \(u(x,0)=u^ 0(x)\), using a finite element process for the space discretization and an implicit Runge-Kutta method for the time integration. In spite of their excellent stability properties and the high orders of accuracy of which they are capable, implicit Runge-Kutta methods exhibit an order reduction phenomenon when applied to stiff problems. This applies both to Gauss and Radau IIA methods and to diagonally implicit methods, where the stage order may be significantly lower than the method order. A detailed analysis of truncation error in the case of the special problem considered here, leads to the conclusion that, for this application, order reduction is avoided.

MSC:
65Z05 Applications to the sciences
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65L20 Stability and convergence of numerical methods for ordinary differential equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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