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ADER schemes are numerical methods, which can reach an arbitrary order ofaccuracy in both space and time. They are based on a reconstruction procedureand the solution of generalized Riemann problems. However, for general boundaryconditions, in particular of Dirichlet type, a lack of accuracy might occur ifa suitable treatment of boundaries conditions is not properly carried out. Inthis work the treatment of Dirichlet boundary conditions for conservation lawsin the context of ADER schemes, is concerned. The solution of generalizedRiemann problems at the extremes of the computational domain, provides thecorrect influence of boundaries. The reconstruction procedure, for data near tothe boundaries, demands for information outside the computational domain, whichis carried out in terms of ghost cells, which are provided by using thenumerical solution of auxiliary problems. These auxiliary problems arehyperbolic and they are constructed from the conservation laws and theinformation at boundaries, which may be partially or totally known in terms ofprescribed functions. The evolution of these problems, unlike to the usualmanner, is done in space rather than in time due to that these problems arenamed here, {\it reverse problems}. The methodology can be considered as anumerical counterpart of the inverse Lax-Wendroff procedure for filling ghostcells. However, the use of Taylor series expansions, as well as, Lax-Wendroffprocedure, are avoided. For the scalar case is shown that the present procedurepreserve the accuracy of the scheme which is reinforced with some numericalresults. Expected orders of accuracy for solving conservation laws by using theproposed strategy at boundaries, are obtained up to fifth-order in both spaceand time.

A strategy to implement Dirichlet boundary conditions in the context of ADER finite volume schemes. One-dimensional conservation laws