The Convergence Rate of Approximate Solutions for Nonlinear Scalar Conservation Laws

Let \ v^ε (x,t) \ε > 0 be a family of approximate solutions for the nonlinear scalar conservation law uₜ + f(u)ₓ = 0 with C₀¹-initial data. Assume that \ v^ε (x,t) \ are Lip^ +-stable in the sense that they satisfy Oleinik’s E-entropy condition. It is shown that if these approximate solutions are Lip'-consistent, i.e., if \| v^ε ( · ,0) - u( · ,0) \|_Lip'(x) + \| vₜ^ε + f(v^ε )ₓ \|_Lip'(x,t) = O(ε ), then they converge to the entropy solution, and the convergence rate estimate \| v^ε ( · ,t) - u( · ,t) \|_Lip'(x) = O(ε ) holds. Consequently, the familiar Lᵖ-type and new pointwise error estimates are derived. These convergence rate results are demonstrated in the context of entropy satisfying finite-difference and Glimm’s schemes.