Sesión Análisis Numérico y OptimizaciónA convergent numerical scheme for the porous medium equation with fractional pressure
Félix del Teso
Universidad Autónoma de Madrid, España - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
We introduce and analyze a numerical approximation of the porous medium equation with fractional potential pressure introduced by Caffarelli and V\'azquez: \[ \partial_t u = \nabla \cdot (u^{m-1}\nabla (-\Delta)^{-\sigma}u) \qquad \text{for} \qquad m\geq2 \quad \text{and} \quad \sigma\in(0,1). \] Our scheme is for one space dimension and positive solutions $u$. It consists of solving numerically the equation satisfied by $v(x,t)=\int_{-\infty}^xu(y,t)dy$, the quasilinear nondivergence form equation \[ \partial_t v= -|\partial_x v|^{m-1} (- \Delta)^{s} v \qquad \text{where} \qquad s=1-\sigma, \] and then computing $u=v_x$ by numerical differentiation. Using upwinding ideas in a novel way, we construct a new and simple, monotone and $L^\infty$-stable, approximation for the $v$-equation. The full scheme then becomes a conservative up-wind finite volume approximation for the $u$-equation. We show local uniform convergence to the unique discontinuous viscosity solution for the $v$-problem, and using ideas from probability theory, we prove that the approximation of $u$ converges up to normalization in $C(0,T; P(\mathbb{R}))$ where $P(\mathbb{R})$ is the space of probability measures under the Rubinstein-Kantorovich (bounded Lipschitz) metric. The analysis includes also fundamental solutions where the initial data for $u$ is a Dirac mass.
Trabajo en conjunto con: Espen R. Jakobsen (Norwegian University of Science and Technology).