This article proposes a new method for data assimilation and data reconciliation problems applicable to systems modeled by conservation laws. The problem is solved directly in the equivalent format of a Hamilton-Jacobi partial differential equation, for which the solution is fully characterized by a Lax-Hopf formula. Using properties of the solution, we prove that when the data of the problem is prescribed in piecewise affine form, the resulting constraints which consist of the partial differential equation in data assimilation and reconciliation problems are convex, and can be instantiated explicitly. This property enables us to identify a class of data assimilation and data reconciliation problems that can be formulated using convex programs in standard form. We illustrate the capabilities of the method for reconstruction of highway traffic flow using experimental data generated from the Mobile Century experiment.