An implicit compact scheme solver with application to chemically reacting flows (Q1777093)

The paper intends to describe a new stable implicit compact scheme solver on non-uniform grids for 1D configurations. The first part of the paper describes the compact scheme discretizations of the first and second-order derivatives of a smooth function on \([ x_{1},x_{N}] \) as \( u^{\prime }=C_{x}u\) or \(u^{\prime \prime }=C_{xx}u\), where \(C_{x}\) and \( C_{xx}\) are dense matrices called stencil matrices. Motivated by the consideration of a nonlinear problem \(f( u,u^{\prime }) =0\), the authors introduce the notion of Jacobian matrix entry which involves \(C_{x}\). The authors then consider the evolution equations \(\partial u/\partial t+c\partial u/\partial x=0\) or \(\partial u/\partial t-\mu \partial ^{2}u/\partial x^{2}=0\). A discrete implicit Euler scheme involving \(C_{x}\) or \(C_{xx}\) is introduced and the authors discuss its stability through a spectral analysis. The main part of the paper deals with the equation \[ \partial u/\partial t+w\partial u/\partial x-\mu \partial ^{2}u/\partial x^{2}+g( x,u,t) =0, \] which covers both the linear and the nonlinear Burgers equations. The authors describe its resolution controlling the time step through a uniform CFL condition and introducing some possible implicit methods: first-order Euler, Crank-Nicolson, or second-order backward difference methods. At each time step, the equation is solved using Briley-McDonald approach for the two first ones and through Newton's solver for the third one. The paper presents several computational results both for the steady-state and for the time-dependent Burgers equations, on uniform and on non-uniform grids. The paper ends with a short description of the ability of such a method in order to solve a stiff problem, namely that of an unstable flame propagation.