Numerical Solution of linear drift-diffusion and pure drift equations on one-dimensional graphs

Abstract

We propose numerical schemes for the approximate solution of problems defined on the edges of a one-dimensional graph. In particular, we consider linear transport and a drift-diffusion equations, and discretize them by extending Finite Volume schemes with upwind flux to domains presenting bifurcation nodes with an arbitrary number of incoming and outgoing edges, and implicit time discretization. We show that the discrete problems admit positive unique solutions, and we test the methods on the intricate geometry of an electrical treeing.

Figures (4)

• Figure 1: Simple one-dimensional domain with three edges and one bifurcation node $v_2$. If we impose homogeneous Neumann conditions on the set of sinks $\mathcal{N}_e = \{v_3,\ v_4\}$ the graph should be extended by also including the nodes $v_5$ and $v_6$ and the additional dashed edeges, according to banasiak2014asymptotic.
• Figure 2: Simple one-dimensional domain with three edges and one bifurcation node $v_1$. The set of edges is $\mathcal{E}=\{e_1=AI,e_2=IB,e_3=IC\}$. We set Dirichlet boundary conditions on $A$ and Neumann boundary conditions on the set of end nodes $\mathcal{N}_e=\{B,C\}$. The only bifurcation node is $I$.
• Figure 3: Segment discretized by contiguous elements without bifurcations. The set of edges is $\mathcal{E}=\{e_1 = AB, \ e_2=BC\}$ and there are no bifurcations. We impose Dirichlet boundary condition on the source node $v_0$ and homogeneous Neumann on the sink $v_2$.
• Figure :

Theorems & Definitions (21)

• Remark 1
• Definition 1
• Definition 2
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Remark 2
• proof