Abrupt phenomena in modelling real-world systems indicate the importance of investigating systems with steep gradients. However, it is difficult to Solve Such systems either analytically or numerically. In 1993. Koren developed a high-resolution numerical computing scheme to deal with compressible fluid dynamics with Dirichlet boundary condition. Recently. Qamar adapted this scheme to numerically solve population balance equations without diffusion terms. This paper extends Koren's scheme for partial differential equations (PDEs) that describe both nonlinear propagation and diffusive effects. and for PDEs with Cauchy or Neumann boundary condition. Accurate and convergent numerical solutions to the test problems have been obtained. The new results are also compared to those obtained by wavelet-based methods. It is shown that the method developed in this paper is more efficient. (c) 2008 Elsevier Ltd. All rights reserved.