An Improved Coarse-Mesh Nodal Integral Method for Partial Differential Equations

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An improved variation of the nodal integral method to solve partial differential equations has been developed and implemented. Rather than treating all of the nonlinear terms as the so-called pseudo-source terms (to be approximated), in this modified version of the nodal integral method, by approximating part of the nonlinear terms in terms of the discrete variable(s) that ultimately result at the end of the formulation process, some or all of the nonlinear terms are kept on the left-hand side in the transverse-integrated equations, which are to be solved analytically. Application of the method to solve the Burgers equation leads to exponential variation within the nodes and shows that the resulting scheme has inherent upwinding. Reconstruction of node interior solution - as a function of one independent variable, and averaged in all others - makes it possible to obtain rather accurate solutions even on a fine scale. Results of the numerical analysis and comparison with results of other methods reported in the literature show that the new method is comparable and sometimes better in accuracy than the currently used schemes. Extension to multidimensional problems is straightforward.

Original languageEnglish (US)
Pages (from-to)113-145
Number of pages33
JournalNumerical Methods for Partial Differential Equations
Issue number2
StatePublished - Mar 1997


  • Burgers equation
  • Coarse-mesh
  • Inherent upwinding
  • Nonlinear PDEs

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


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