December 9, 2022 is the 90th birthday of Tong Zhang, a mathematician in Institute of Mathematics, Chinese Academy of Sciences where he was always working on the Riemann problem for gas dynamics in his mathematical life. To celebrate his 90th birthday and great contributions to this specific field, we organize this focused issue in the journal Communications on Applied Mathematics and Computation, since the Riemann problem has been proven to be a building block in all fields of theory, numerics and applications of hyperbolic conservation laws.

Professor Zhang began his journey on the Riemann problem motivated by Gelfand’s paper on the theory of quasilinear equations soon after getting his bachelor’s degree from Sichuan University in 1956. His earlier influential work may be the one with Yu-fa Guo about an initial value problem for compressible Euler equations whose solution consists of a sequence of Riemann solutions and their interactions. This work and other related contributions are subject to a special class of “Chinese initial data” named by western mathematicians such as Peter Lax and Joel Smoller. His book “The Riemann Problem and Interaction of Waves in Gas Dynamics” in 1989, co-authored with Professor Ling Hsiao, collected his works with collaborators during 1963–1986, commented by AMS book reviews as a valuable supplement or the sequel to Courant and Friedrichs’s book “Supersonic Flows and Shock Waves”.

It seemed natural to work on two-dimensional (2-D) problems after his good experience on one-dimensional (1-D) ones. Also, 2-D wave phenomena in gas dynamics such as various shock reflections are very rich. However, it was not easy as a pioneering contribution at that time, still looks creative and audacious even today, to provide a doable mathematical formulation without loss of 2-D inherent wave configurations. His four-quadrant 2-D Riemann problem with Yuxi Zheng defines this direction and has become the benchmark to verify numerical performances, although theoretical progress is on-going relatively slowly by his students and successors. His book on the 2-D Riemann problem with Jiequan Li et al., received an AMS review saying his research group as a “Chinese School of Mathematics”.

The study of the Riemann problem is, of course, not just a purely mathematical interest, but to disclose the elementary wave patterns. Beyond the gas dynamics, many other non-classical wave patterns are awaiting for discovery. The delta shock wave, a family of singular shocks discovered independently with his former student Dechun Tan, was a branch-new concept. Its rigorous analysis was gradually being made in progress. The album in Memoirs of AMS “The Riemann Problem for the Transportation Equations in Gas Dynamics” with Wancheng Sheng recorded his thinking on how to rigorize a new concept and relate it to physics. Other works on gas combustions are full of wisdom on global entropy conditions of detonation and deflagration, in sharp contrast with classical entropy conditions of shocks.

The contributors to this issue are more or less benefited from or influenced by his contribution.