In this paper, we adapt a WENO limiter [45] to the CPR framework solving hyper-bolic conservation laws to make it more robust for shocked flows and uniformly high order accurate. Also, we extend the positivity-preserving limiter in [43, 44, 36] to the CPR framework. On each time level, we first use the WENO limiter to reconstruct the solution polynomials on those troubled cells, then use the positivity-preserving limiter to modify the solution polynomials in each cell if necessary. Finally, we update the nu-merical values at the solution points, and perform the normal CPR procedure to march to the next time level. Since this WENO limiter uses information only from immediate neighbors, it is very simple to implement and can maintain the compactness of the CPR framework. Also, we only perform this WENO limiter on the solution polynomials which can be discontinuous among adjacent cells, thus the conservativeness of the CPR frame-work will not be harmed. Numerical results in one and two dimensions are provided to show that this WENO limiting procedure can simultaneously maintain uniform high order accuracy of the CPR framework in smooth regions and control spurious numerical oscillations near discontinuities. In future work we will extend the WENO limiter to CPR framework on unstructured meshes along the lines of [47].
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