First-, second-, and third-order finite-volume schemes for diffusion

In this paper, we present constructions of first-, second-, and third-order schemes for diffusion by the method introduced in Nishikawa (2007) [10]. In this method, numerical schemes for diffusion are constructed by advection schemes via an equivalent hyperbolic system. This paper demonstrates that the method enables straightforward constructions of diffusion schemes for finite-volume methods on unstructured grids. In particular, it is demonstrated that a robust first-order upwind scheme leads to a robust first-order diffusion scheme, and a high-order advection scheme leads to a high-order diffusion scheme. It is shown that first-, second-, and third-order schemes are capable of producing first-, second-, and third-order accurate solution gradients, respectively, on irregular grids. Accuracy, Fourier stability, and the energy stability of the developed schemes are discussed. A new hyperbolic diffusion system having virtually no source terms is also introduced to simplify the construction of the third-order scheme. Numerical results are presented for regular and irregular triangular grids to demonstrate not only the superior accuracy but also the accelerated steady convergence over a traditional method.