Space-Time Adaptive Solution of First Order PDES

An explicit time-stepping method is developed for adaptive solution of time-dependent partial differential equations with first order derivatives. The space is partitioned into blocks and the grid is refined and coarsened in these blocks. The equations are integrated in time by a Runge-KuttaFehlberg method. The local errors in space and time are estimated and the time and space steps are determined by these estimates. The error equation is integrated to obtain global errors of the solution. The method is shown to be stable if one-sided space discretizations are used. Examples such as the wave equation, Burgers’ equation, and the Euler equations in one space dimension with discontinuous solutions illustrate the method.