Extend numerical-smoothness analysis to multidimensional and higher-order hyperbolic problems

Extend the numerical-smoothness error-analysis framework from one-dimensional scalar first-order hyperbolic equations to two-dimensional scalar equations, hyperbolic systems, and second-order hyperbolic problems.

Background

The paper reports numerical-smoothness analyses for one-dimensional scalar conservation laws, including discontinuous Galerkin and WENO methods and, in one case, shock tracking. The stated framework relies on local spatial error analysis together with exact error propagation, such as the L1L_1 contraction property for scalar conservation laws.

The notes explicitly identify the multidimensional, system-valued, and second-order hyperbolic settings as unresolved directions. Extending the theory would require handling more complicated wave interactions, derivative jumps, shocks, contact discontinuities, and rarefaction structures.

References

For first order hyperbolic problems, research has only been done for 1-D scalar equations. 2-D scalar equations and hyperbolic systems are still to be studied. Second order hyperbolic problems still need to be studied.

— Numerical Smoothness and Error Analysis for Evolution Equations  (2610.12888 - Sun, 9 Oct 2026) in Section 4, “Future research”