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Article Contents

# A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation

• In this paper we prove the convergence of a Crank-Nicolson type Galerkin finite element scheme for the initial value problem associated to the Benjamin-Ono equation. The proof is based on a recent result for a similar discrete scheme for the Korteweg-de Vries equation and utilizes a local smoothing effect to bound the $H^{1/2}$-norm of the approximations locally. This enables us to show that the scheme converges strongly in $L^{2}(0,T;L^{2}_{\text{loc}}(\mathbb{R}))$ to a weak solution of the equation for initial data in $L^{2}(\mathbb{R})$ and some $T > 0$. Finally we illustrate the method with some numerical examples.

Mathematics Subject Classification: Primary: 35Q53, 65M60; Secondary: 35Q51, 65M12.

 Citation:

• Figure 1.  Numerical approximation for $N = 256$ and exact solution for $t = 0$, $90$ and $180$, respectively positioned from left to right in the plot, for initial data $u_{s2}$

Table 1.  Relative $L^{2}$-error, $I_1$, $I_2$ and $I_3$ at $t = 90$ and $t = 180$ for initial data $u_{s2}$

 t N E rateE I1 I2 I3 90 128 0.01844 -1.451.580.681.160.08 3.79×l0-5 -7.05×l0-4 5.86×l0-3 256 0.05021 -6.61×l0-6 -3.85×l0-3 1.65×l0-2 512 0.01678 -6.64×l0-6 -9.96×l0-4 4.70×l0-3 1024 0.01044 -4.64×l0-6 3.62×l0-4 -1.57×l0-3 2048 0.00467 -3.25×l0-6 4.16×l0-5 -5.71×l0-6 4096 0.00442 -2.29×l0-6 4.12×l0-6 2.07×l0-4 180 128 0.11959 -1.321.750.742.350.89 1.57×l0-4 -6.45×10-4 -1.56×l0-2 256 0.29755 2.48×l0-6 -7.80×l0-3 3.32×l0-2 512 0.08869 -3.76×l0-6 -2.42×l0-3 1.12×l0-2 1024 0.05295 -2.69×10-6 9.22×l0-4 -4.11×10-3 2048 0.01040 -1.82×10-6 1.16×10-4 -4.74×10-4 4096 0.00561 -1.26×l0-6 1.46×10-5 -1.67×l0-5
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