The pricing of Asian options is explored by numerically solving the associated partial differential equations. The authors demonstrate that numerical PDE techniques commonly used in finance for standard options are inaccurate in the case of Asian options and illustrate modifications which alleviate this problem. In particular, the usual methods generally produce solutions containing spurious oscillations. Flux-limiting techniques originally developed in the field of computational fluid dynamics are adapted in order to rapidly obtain accurate solutions. It is shown that flux-limiting methods are total variation diminishing (and hence free of spurious oscillations) for nonconservative PDEs such as those typically encountered in finance, for fully explicit, and fully and partially implicit, schemes. The van Leer flux limiter is also modified so that the second-order total variation diminishing property is preserved for nonuniform grid spacing.