Backward error analysis of a full discretization scheme for a class of semilinear parabolic partial differential equations (Q1863471)

The semilinear partial differential equations (1) \( \frac{\partial u}{\partial t}-D \triangle u=F(u(t,x)) \) where \[ u=(u_1,u_2,\ldots, u_n), \] \(D=\text{diag}(d_1,\ldots, d_n)\) with \(d_i>0\), and \(F=(f_1,f_2,\ldots, f_n)\) with periodic boundary conditions on \(\Omega=[0, 2\pi]^d\), \(d=1, 2, 3\) and initial conditions \(u(0)=u_0\) in the Sobolev space \(H^S_{\text{per}}(\Omega, \mathbb{R}^n) \subset C^0 (\Omega, \mathbb{R}^n)\), i.e. \(s>d/2\) as a model a system of reaction-diffusion equations is considered. For a definition of \(H_{\text{per}}^s(\Omega, \mathbb{R}^n)\) for nonintegers see [\textit{R. Teman}, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, Vol. 68, Springer, New York (1988; Zbl 0662.35001)]. The nonlinearity F is assumed to be an entire function \(\mathbb{R}^n \to \mathbb{R}^n\). Equation (1) defines a semiflow on \(H^S_{\text{per}}(\Omega, \mathbb{R}^n)= D(A^{s/2})\) with the differential operator \(A=-D\triangle\). A numerical method for the above-mentioned reaction-diffusion equations with analytic nonlinearity is presented, for which a rigorous backward error analysis is possible. It is constructed a modified equation, which describes the behaviour of the full discretization scheme up to exponentially small errors in the step size. In the construction the numerical scheme is first exactly embedded into a nonautonomous equation. This equation is then averaged with only exponentially small remainder terms. The long-time behaviour near hyperbolic equilibrium, the persistence of homoclinic orbits and regularity properties are analyzed.