An adaptive discontinuous Galerkin method for very stiff systems of ordinary differential equations

摘要

We present a discontinuous Galerkin (DG) method for the numerical solution of stiff systems of ordinary differential equations (ODEs). We use a standard DG variational formulation with polynomials of degree k in each time interval. We show that the method is A-stable for every k. We then introduce a hierarchical Legendre finite element basis and we show that a whole family of approximations can be obtained simply by truncating the last p degrees of freedom from the computed solution. We show that these approximations converge to order k+1−p in L2-norm and to order k+1/2−p in supremum norm. We then show how this can be used to control the error and the time step length. We present numerical examples of solutions on very stiff problems and on stiff problems with very long time integration where the time step length can vary on many orders of magnitude.