On Some Adaptive Numerical Methods for Linear Advection Equations

Abstract

The most important feature of adaptive numerical methods can be characterized as their capability to concentrate computational efforts where they are most needed. As usual, in adaptive methods redistribution of the efforts happens automatically in dynamic manner with account of properties of derived in the process of computations approximate solution of the problem under consideration. Such flexibility allow adaptive methods to achieve comparable accuracy and at the same time substantially to reduce computational costs in comparison with non adaptive ones. Example of such successful adaptive method is given in [7]. Namely, for calculation of steady state flow over aircraft’s bow shape in 2D comparable numerical results are achieved by adaptive numerical method with 700-1000 equations and by non adaptive one with 100 000 equations. Because of such significant advantage development of good adaptive numerical methods is and will be important despite increasing the power of computers. The following 3 basic approaches for achieving adaptation are often used: