Self-similar structure of the entry and exit discontinuities in two-way diffusion equations

Two-way diffusion equations arising in kinetic problems relating to electron scattering and in Brownian particle dynamics present singularities absent from conventional diffusion equations. Although calculations by Stein & Bernstein, and Fisch & Kruskal have revealed the formation of entry and exit slope discontinuities at the critical points where the velocity changes sign, the analytical structure of these discontinuities remains unclear. Here we fill this gap via a local similarity variable analysis, illustrated through the two-way diffusion equation $y \partial n/\partial x=\partial ^2 n/\partial y^2$ in $-1 \leq y \leq 1$ ; $0 \leq x \leq L$ , with $n(x,\pm 1)=0$ with various entry conditions $n(0,y)_{y\gt 0}$ , and the exit condition $n(L,y)_{y\lt 0}=0$ . The similarity variable $\eta =y/x^{1/3}$ permits the analytical characterization of the entry discontinuity, except for constants determined by matching with numerical solutions obtained with two numerical schemes: separation of variables following the construction of Beals, or finite-difference discretization of the transient partial differential equation, which converges in time to a solution almost identical to the separation of variables solution. Although the slope discontinuity depends markedly on the initial condition $n(0,y)_{y\gt 0}$ , a simple general similarity solution structure emerges empirically, always involving a spontaneous singular contribution $C |y|^{1/2}$ at $x=0,y\lt 0$ . Slow convergence of both numerical solutions near $\{x,y\}=\{0,0\}$ is attributed to the poor eigenfunction representation of the ever-present singular solution component $|y|^{1/2}$ . The similarity approach applies equally to other two-way diffusion equations when the coefficient of $\partial n/\partial x$ changes sign linearly with $y$ . It can also be extended to situations where this coefficient is discontinuous at the critical points.