In this thesis, we construct central (Godunov-type) schemes for hyperbolic conservation laws and Hamilton-Jacobi (H-J) equations, and study the {dollar}Lsp{lcub}1{rcub}{dollar}-convergence theory for general approximate solutions to Cauchy problems for H-J equations.; We construct high-resolution, non-oscillatory, non-staggered central schemes for conservation laws. Two mechanisms are provided: re-averaging over a high- order reconstruction and dimension-wise splitting on multi-dimensional conservation laws. For H-J equations, we construct new (first- and second-order) Godunov-type schemes which possess a global interpolant (projection operator). In addition, we prove that the {dollar}Lsp{lcub}1{rcub}{dollar}-error of general approximate H-J solutions which satisfy the semi-concave stability condition can be controlled by their initial and perturbed errors. When applied to viscous regularization and first-order Godunov-type schemes constructed above, our {dollar}Lsp{lcub}1{rcub}{dollar}-theory implies a convergence rate of order one.
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