Explicit and implicit time integration algorithms for the two-dimensio
nal Euler equations on unstructured grids are presented. Both cell-cen
tered and cell-vertex finite volume upwind schemes utilizing Roe's app
roximate Riemann solver are developed. For the cell-vertex scheme, a f
our-stage Runge-Kutta time integration, a four-stage Runge-Kutta time
integration with implicit residual averaging, a point Jacobi method, a
symmetric point Gauss-Seidel method, and two methods utilizing precon
ditioned sparse matrix solvers are presented. For the cell-centered sc
heme, a Runge-Kutta scheme, an implicit tridiagonal relaxation scheme
modeled after line Gauss-Seidel, a fully implicit lower-upper (LU) dec
omposition, and a hybrid scheme utilizing both Runge-Kutta and LU meth
ods are presented. A reverse Cuthill-McKee renumbering scheme is emplo
yed for the direct solver to decrease CPU time by reducing the fill of
the Jacobian matrix. A comparison of the various time integration sch
emes is made for both first-order and higher order accurate solutions
using several mesh sizes. higher order accuracy is achieved by using m
ultidimensional monotone linear reconstruction procedures. The results
obtained for a transonic flow over a circular arc suggest that the pr
econditioned sparse matrix solvers perform better than the other metho
ds as the number of elements in the mesh increases.