IDeC for Inhomogeneous Problems

For inhomogeneous equations the method of iterated defect correction can be ap­plied in two different ways, which leads both to the same corrected solution but with different time efficiencies. The difference is even bigger for source terms depending on the solution itself. In the following work we will present both inhomogeneous equations with source terms depending only on space and source terms including the solution itself. To describe the different formulations we take the scalar evolu­tion equation

ut + f (u)x = s(u). (16)

again in 1D as example with the source term s depending on the solution u. If we apply the iterated defect correction on this problem as done before, computing the local defect only in the flux terms, we can write the modified equation (5) as

uf+11+ f (uk+1])x = s(w) – (f(w[% – f (U%) . (17)

‘———— V———– ‘

dk+1]

The integration of each term is done as shown above what leads to a similar semi discrete representation of the modified equation (5) with the additional integral

tn+Atxi+1/2

J J s(wi)dxdt,

tn xi-1/2

of the source term s in the cell i. To achieve the consistency order of the reconstruc­tion in the iterated defect correction procedure with the above formulation (17), it is important to compute the source term with the high order accuracy. This implies a reconstruction in each iteration of the basic method and in 2D and 3D an integration with much more Gauss points than used for the basic scheme of lower order is ne­cessary. The high computational cost can be reduced by reformulating the problem in equation (17). Instead of taking only the fluxes into account for the local defect, we propose to include the source term as well in the definition of the local defect. This yields

uf+1] + f (uk+1] )x = s(u[k+1) – [f (w[k )x – s(wk ) – (f (uh )x – s(u[k] )

dk+1]

a new modified equation where the source term in the iteration of the basic scheme is now integrated with the lower order accuracy whereas the reconstruction of the source term is only done once per defect iteration to compute the local defect.