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An improved compact finite difference scheme for solving an N‐carrier system with Neumann boundary conditions
Author(s) -
Dai Weizhong
Publication year - 2011
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20531
Subject(s) - mathematics , compact finite difference , partial differential equation , partial derivative , neumann boundary condition , boundary value problem , boundary (topology) , von neumann stability analysis , scheme (mathematics) , mathematical analysis , convergence (economics) , stability (learning theory) , computer science , economics , economic growth , machine learning
Recently, we have developed a higher‐order and unconditionally stable compact finite difference scheme for solving a model of energy exchanges in an N‐carrier system with Neumann boundary conditions, which extends the concept of the well‐known parabolic two‐step model for microheat transfer. However, the combined compact finite difference scheme for the boundary is second‐order accurate. Unfortunately, our statement in (Dai and Tzou, Numer Methods Partial Differential Equation, Zhao et al., Numer Methods Partial Differential Equations 23 (2007), 949–959.), that it is a fourth‐order scheme is inaccurate, because the scheme was multiplied by Δ x 2 in the derivation. In this article, we develop a new combined compact finite difference scheme for the boundary, which is third‐order accurate. Using the exact same proof for stability analysis as in (Dai and Tzou, Numer Methods Partial Differential Equations), the new scheme is unconditionally stable with respect to the initial conditions and source terms. The improved compact scheme is then tested by a numerical example. Results show that the convergence rate with respect to the spatial variable from the new scheme is higher and the solution is much more accurate, when compared with those obtained using our previous compact scheme in (Dai and Tzou, Numer Methods Partial Differential Equations). © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011