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Methods for the numerical solution of the Benjamin-Bona-Mahony-Burgers equation. (English) Zbl 1165.65059

The authors study the numerical solution of the Benjamin-Bona-Mahony-Burgers (BBMB) equation
\[ u_t-(\varphi(x,t)u_{xt})_x-\alpha u_{xx}+f_x(u)=0, \quad x \in \Omega, \quad t \in (0,T], \] with boundary and initial condition \[ u(x,t)=0, \quad x \in \partial\Omega, \quad t \in (0,T], \qquad u(x,0)=u_0(x), \quad x \in \Omega, \] where \(f(u)= u^2/2+u, \alpha>0, \Omega=(0,1)\) and \(T>0\). Moreover it is assumed that there exists constants \(M, M_1\) and \(M_2\) such that
\[ 0<M_1 \leq \varphi(x,t) \leq M_2 \text{ and } |\varphi_t(x,t)| \leq M, \quad x \in \Omega, \quad t \in [0,T]. \]
For \(\alpha=0\) and \(\varphi=1\) one has the regularized long-wave equation (without dissipation), which is also known as Benjamin-Bona-Mahony equation. The dissipation term \(\alpha u_{xx}\) is the same as in the Burgers equation.
The existence of a solution of the given problem in a weak formulation and its boundedness in the space \(L^\infty(H_0^1(\Omega))\) are shown with the aid of Galerkin’s method. The weak problem is semi-discretized with \(H^1_0(\Omega)\)-finite elements of order \(r\) and quasi-optimal convergence order is proved. Fully discrete approximations are obtained using the backward differentiation method of order one and two. The proved error estimates are quasi-optimal in time and space.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin 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
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35Q30 Navier-Stokes equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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References:

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