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Conservative, high-order numerical schemes for the generalized Korteweg- de Vries equation. (English) Zbl 0824.65095
A class of fully-discrete schemes for the numerical solution of the problem \[ u_ t + u^ p u_ x + u_{xxx}= 0,\quad u(x,0) = u_ 0(x),\quad 0 \leq x \leq 1,\tag{1} \] \(u_ 0\) belonging to a suitable class of periodic functions with period 1, is analysed, implemented and tested. In fact, the schemes are written for (1) posed in a slightly more general form \[ v_ t + \eta \cdot v_ x + v^ p \cdot v_ x + \varepsilon v_{xxx} = 0\quad v(x,0) = u_ 0(x).\tag{2} \] Problem (2) is equivalent to (1) by means of variable change \(v(x,t) = u(\beta(x - \eta t), \beta t)\), \(\beta = \varepsilon^{-1/2}\).
The spatial discretizations are effected using smooth splines of quadratic or higher degree, and the temporal discretizations are conservative, multistage, implicit Runge-Kutta methods. A proof is presented showing convergence of the numerical approximations to the true solution of the initial value problem in the limit of vanishing spatial and temporal discretization. Numerical experiments are done for the investigation of the instability of the solitary-wave solutions of (2).
Reviewer: K.Zlateva (Russe)

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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