| In this dissertation we introduce a class of high order Godunov-type schemes that are based on quadratic cell interface values, and we prove that they are TVD under a suitable CFL restriction. The choices of quadratic interface values are similar to those of the ENO reconstruction, but are subtly modified to yield a TVD correction. Moreover, we establish that our TVD procedure is third order accurate generically. To achieve better resolution of contact discontinuities, we introduce a TVD contact sharpening procedure, which is tuning parameter independent, sufficiently compressive, and easy to extend to multi-dimension problems. We prove TVD stability and accuracy for this new approach. In the multi-dimension case, we introduce and analyse an unsplit extension of our one dimensional techniques. The key feature of our unsplit scheme is that it is essentially devoid of any grid orientation effects. Finally, we extend this scheme to nonlinear hyperbolic systems and demonstrate its effectiveness on a wide variety of benchmark test problems. |
