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As per the https://www.sciencedirect.com/science/article/pii/S0021999197957934 (original paper), multiplying the Jacobian with \kappa = 0.5 is not right. It is leading to non-convergence.
Further, dpStar_dU[iVar] calculations need to be relooked. As per the original paper, density should be at "j" when it is being used at "i". Though I have not seen a major difference, but currently testing the same more for strong shock cases.