WHY NONCONSERVATIVE SCHEMES CONVERGE TO WRONG SOLUTIONS - ERROR ANALYSIS

This paper attempts to give a qualitative and quantitative description of the numerical error introduced by using finite difference schemes in nonconservative form for scalar conservation laws. We show that the se schemes converge strongly in L(loc)1 norm to the solution of an inh omogeneous conservation law containing a Borel measure source term. Mo reover, we analyze the properties of this Borel measure, and derive a sharp estimate for the L1 error between the limit function given by th e scheme and the correct solution. In general, the measure source term is of the order of the entropy dissipation measure associated with th e scheme. In certain cases, the error can be small for short times, wh ich makes it difficult to detect numerically. But generically, such an error will grow in time, and this would lead to a large error for lar ge-time calculations. Finally, we show that a local correction of any high-order accurate scheme in nonconservative form is sufficient to en sure its convergence to the correct solution.