TABLE OF CONTENTS The successful high-resolution methods discussed in earlier chapters are predominantly second-order in formal accuracy. As one might guess there have been focused efforts to extend these methods to higher order accuracy. In this chapter we will introduce several approaches to achieving this end.
The development of accurate and efficient numerical methods for time-dependent transport phenomena, both for practical applications as well as for fundamental studies, remains a challenging task for numerical analysts and practitioners. This chapter is concerned with the development and application of numerical methods of very high-order of accuracy; in the context of this chapter this means at least third and possibly fourth, fifth or higher. Note that several developments discussed in this chapter are applicable to the broader field of computational physics. Obvious advantages of high-order methods are: [1]
The desired accuracy is attained with reduced memory and storage requirements. This is of paramount importance for any three-dimensional time dependent problems as well as for multiphysics problems involving many equations, such as in multiphase flows.
In conjunction with current computing resources available, very high-order methods may be the only way of providing reliable solutions for practical engineering problems, which are presently studied using under-resolved simulations, i.e., using coarse computational grids. Low-order methods, first or second-order, can be completely inadequate without resorting to very fine meshes that are not affordable with current computers.
High-order methods may offer higher accuracy in fundamental studies concerned with the modeling of associated processes not yet fully understood, for example, turbulence.
