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 these schemes converge strongly in $L_{\operatorname {loc}}^1$ norm to the solution of an inhomogeneous conservation law containing a Borel measure source term. Moreover, we analyze the properties of this Borel measure, and derive a sharp estimate for the ${L^1}$ error between the limit function given by the scheme and the correct solution. In general, the measure source term is of the order of the entropy dissipation measure associated with the scheme. In certain cases, the error can be small for short times, which makes it difficult to detect numerically. But generically, such an error will grow in time, and this would lead to a large error for large-time calculations. Finally, we show that a local correction of any high-order accurate scheme in nonconservative form is sufficient to ensure its convergence to the correct solution.