| The reliable simulation of shockwaves is critical in the prediction and study of many phenomena, where abrupt changes in material properties due to shockwaves can greatly affect regions of interest and activate physical mechanisms. When a physical shockwave is formed, it moves through the flow with a certain speed, having some finite width determined by physical dissipation until it encounters some event in its path. For numerical shockwaves, however, a numerical width is enforced, often much greater than the physical width. With this numerical width comes the formation of intermediate states having no direct physical interpretation. Even as the mesh is refined, these intermediate states do not go away; they simply occupy less space. The existence of intermediate states does raise some doubt, however, about how closely a captured shockwave may emulate an ideal discontinuous shockwave, or a real physical one.;There are in fact several types of error associated with intermediate shock states such as errors in shock position, spurious waves, or unstable shock behavior. These errors can be classified as numerical shockwave anomalies; they are numerical artifacts formed due to the presence of captured shockwaves within the flow solution. Each numerical shockwave anomaly is directly related to the nonlinearity of the jump conditions and to a resulting ambiguity in sub-cell shock position in a stationary shock.;Two new flux functions are developed that do not have this ambiguity. On all of the shock anomalies in one-dimension, both flux functions show improvement on existing methods without smearing or diffusing the shock. They are also shown to perform adequately within a second-order framework and on two-dimensional problems, important for the practicality of the method. While they are still susceptible to many of the problems that occur in Roe's Riemann solver and several other known issues, these methods serve to validate the philosophy and approach taken in this thesis: by enforcing a linear shock structure and unambiguous sub-cell shock position, numerical shockwave anomalies are dramatically reduced. |
