Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a ighted W1,1 space. Convergence rate for the derivative of the approximate solutions is established under the assumption that a ak pointwise-error estimate is given. In other words,  are able to convert ak pointwise-error estimates to optimal error bounds in a ighted W1,1 space.#R##N#For convex conservation laws, the assumption of a ak pointwise-error estimate is verified by Tadmor [ SIAM J. Numer. Anal., 28 (1991), pp. 891--906]. Therefore, one immediate application of our W1,1- convergence theory is that for convex conservation laws  indeed have W1,1-error bounds for the approximate solutions to conservation laws. Furthermore, the $\cO(\en)$-pointwise-error estimates of Tadmor and Tang [ SIAM J. Numer. Anal., 36 (1999), pp. 1739--1758] are recovered by the use of the W1,1-convergence result.
['Tao Tang', 'Zhen-Huan Teng']
I am writing a research paper on a idea On the Regularity of Approximate Solutions to Conservation Laws with Piecewise Smooth Solutions
[{"from": "human", "value": "I am writing a research paper on a idea On the Regularity of Approximate Solutions to Conservation Laws with Piecewise Smooth Solutions"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a ighted W1,1 space. Convergence rate for the derivative of the approximate solutions is established under the assumption that a ak pointwise-error estimate is given. In other words,  are able to convert ak pointwise-error estimates to optimal error bounds in a ighted W1,1 space.#R##N#For convex conservation laws, the assumption of a ak pointwise-error estimate is verified by Tadmor [ SIAM J. Numer. Anal., 28 (1991), pp. 891--906]. Therefore, one immediate application of our W1,1- convergence theory is that for convex conservation laws  indeed have W1,1-error bounds for the approximate solutions to conservation laws. Furthermore, the $\\cO(\\en)$-pointwise-error estimates of Tadmor and Tang [ SIAM J. Numer. Anal., 36 (1999), pp. 1739--1758] are recovered by the use of the W1,1-convergence result."}]