On an adaptive time stepping strategy for solving nonlinear diffusion equations (Q1803291)

The following nonlinear diffusion equations are solved by a new time step selection procedure. For an \(r\)-dopant diffusion problem in a silicon medium \(\Omega\), the concentrations of dopants in \(\Omega\) at time \(t\) may be described by \[ \partial C_ k/\partial t = \text{Div}[D_ k\text{ grad }C_ k + Z_ kC_ k\text{ grad }\Phi],\quad k = 1,\dots,r,\tag{1} \] where \(C_ k = C_ k(x,y,t)\) is the concentration for the \(k\)th dopant, \(D_ k\) is the diffusion coefficient, \(Z_ k = 1\) or \(-1\) depends on the dopant and \(\Phi = \Phi(\Sigma Z_ kC_ k)\) is the electrostatic potential. Denote by \(C = \sum^ r_{k=1}Z_ kC_ k\) the total concentration. Then the potential function is calculated by \(\Phi = \log(n/n_ i)\), where \(n = {1\over 2}(C+\sqrt{C^ 2 + 4n^ 2_ i})\) is the electron concentration and \(n_ i\) is the intrinsic electron concentration at the process temperature. The transformation of \(f_ k = \log C_ k\) converts (1) into the system \[ \partial f_ k/\partial t = \text{Div}[D_ k\text{ grad }(f_ k + Z_ k\Phi)] + \text{grad }f_ k \cdot \text{grad}(f_ k + Z_ k\Phi)\quad k = 1,\dots,r.\tag{2} \] For a particular semi-implicit time stepping scheme based on operator splitting, the authors propose a strategy for automatic time step selection. It is based on equidistributing local truncation errors (LTEs) in both time and space discretizations. For finite element methods, the authors show that good approximations to LTEs are always possible so that the strategy can be applied.