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A compact local one‐dimensional scheme for solving a 3D N ‐carrier system with Neumann boundary conditions
Author(s) -
Dai Weizhong
Publication year - 2010
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.20476
Subject(s) - compact finite difference , mathematics , neumann boundary condition , discretization , partial differential equation , von neumann stability analysis , boundary value problem , boundary (topology) , mathematical analysis , convergence (economics) , finite difference method , parabolic partial differential equation , finite difference , scheme (mathematics) , poincaré–steklov operator , robin boundary condition , economics , economic growth
We consider a mathematical model for thermal analysis in a 3D N ‐carrier system with Neumann boundary conditions, which extends the concept of the well‐known parabolic two‐step model for micro heat transfer. To solve numerically the complex system, we first reduce 3D equations in the model to a succession of 1D equations by using the local one‐dimensional (LOD) method. The obtained 1D equations are then solved using a fourth‐order compact finite difference scheme for the interior points and a second‐order combined compact finite difference scheme for the points next to the boundary, so that the Neumann boundary condition can be applied directly without discretizing. By using matrix analysis, the compact LOD scheme is shown to be unconditionally stable. The accuracy of the solution is tested using two numerical examples. Results show that the solutions obtained by the compact LOD finite difference scheme are more accurate than those obtained by a Crank‐Nicholson LOD scheme, and the convergence rate with respect to spatial variables is about 2.6. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010