TABLE OF CONTENTS High-resolution methods were developed specifically for compressible highspeed flows. The most vexing challenge in high speed flows is successfully computing shock waves where all variables are discontinuous. High-resolution methods provided the means to compute shocks without (significant) oscillations while mitigating the dissipative solutions commonly associated with first-order solutions. They also allow higher accuracy of flows away from discontinuities where the flow is "smooth." It is important to understand the algorithmic structure of the foundational methods used in conjunction with high resolution methods. One consideration that is central to these methods is the use of conservation form. This is a consequence of the Lax-Wendroff theorem [321]: If a difference equation is in conservation form and is consistent with the original conservation law as well as stable it will converge to the correct weak solution of that conservation law. This powerful theorem is wise to adhere to because it supplies what little assurances can be made in solving flows with shocks.
If one has a compressible flow solver sometimes the simplest thing to do is run the code at a low Mach number. The downside of using this approach is efficiency. Nonetheless this is common practice because it is often faster to run an inefficient calculation than write an efficient code. In other cases the low speed or incompressible solutions...
