NUMERICAL SOLUTIONS OF CONSERVATION LAWS (TVD SCHEME, TVB, GLOBALLY HIGH ORDER)

In the computation of conservation laws u(,t) + f(u)(,x) = 0, TVD (total-variation-diminishing) schemes have been very successful. TVB (total-variation-bounded) schemes share most of the advantages and may remove some of the disadvantages (e.g. local degeneracy of accuracy at critical points) of TVD schemes. Included in this dissertation are a class of m-step Runge-Kutta type TVD schemes with CFL number equaling m; a procedure to obtain uniformly high order in space TVB schemes; a class of TVD high order time discretizations; a special boundary treatment which keeps the high order of the scheme up to the boundary and preserves the TVB properties in the non-linear scalar and linear system cases; a discrete entropy inequality for a modified Lax-Wendroff scheme applied to Burgers' equation; and discussions about error propagation in large regions.