Exponential Compact Higher Order Scheme for Nonlinear Steady Convection-Diffusion Equations (Communications in Computational Physics, 2011)
This paper presents an exponential compact higher order scheme forConvection-Diffusion Equations (CDE) with variable and nonlinear convection coefficients. The scheme is O(h4) for one-dimensional problems and produces a tri-diagonalsystem of equations which can be solved efficiently using Thomas algorithm. For two-dimensionalproblems, the scheme produces an O(h4+k4) accuracy over a compactnine point stencil which can be solved using any line iterative approach with alternatedirection implicit procedure. The convergence of the iterative procedure is guaranteedas the coefficient matrix of the developed scheme satisfies the conditions required tobe positive. Wave number analysis has been carried out to establish that the scheme iscomparable in accuracy with spectral methods. The higher order accuracy and betterrate of convergence of the developed scheme have been demonstrated by solving numerousmodel problems for one- and two-dimensional CDE, where the solutions havethe sharp gradient at the solution boundary.