Abrupt phenomena in modelling real-world systems indicate theudimportance of investigating systems with deep gradients. However,udit is difficult to solve such systems either analytically orudnumerically. In 1993, Koren developed a high-resolution numericaludcomputing scheme to deal with compressible fluid dynamics withudDirichlet boundary condition. Recently, Qamar adapted this schemeudto numerically solve population balance equations withoutuddiffusion terms. This paper extends Koren's scheme for partialuddifferential equations (PDEs) that describe both nonlinearudpropagation and diffusive effects, and for PDEs with Cauchyudboundary condition. Accurate and convergent numerical solutions toudthe test problems have been obtained. The new results are alsoudcompared to those obtained by wavelet-based methods. It is shown that udthe method developed method in this paper is more efficient.
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