A kernel-based method for parabolic equations with nonlinear convection terms (Q1182664)

The author proposes an algorithm for the numerical solution of the parabolic initial boundary value problem \(u_ t=Lu+V(u)_ x\), \(x\in\Omega\subset\mathbb{R}^ n\); \(u(x,t)=0\), \(x\in\Gamma=\partial\Omega\), \(t>0\); \(u(x,0)=u_ 0(x)\), \(x\in\Omega\), with nonlinear convection term \(V(u)_ x\) and 2nd-order symmetric uniformly elliptic linear differential operator \(L\). The algorithm is based on the representation of the solution \(u(x,t)\) via Green's function and on the time discretization of this representation via the trapezoid rule. For the space discretization, one can use the finite difference method as well as the finite element method. Stability results and error estimates are given. Finally, the author presents some numerical results for some simple spatially one-dimensional problems.