Efficient non-iterative multi-point method for solving the Riemann problem

In computational fluid dynamics (CFD), the Riemann problem is vital for precise fluid flow calculations as it involves the accurate computation of the interactions between different states of fluid flow. Approximate non-iterative Riemann solvers are commonly used for efficiency but introduce errors in complex cases. An exact Riemann solver offers higher accuracy by directly solving the problem without approximations or simplifications. However, it involves nonlinear equations, leading to iterative methods. To promote the use of exact solver, an efficient non-iterative multi-point (NIM) method is proposed to solve the Riemann problem as it has a convergence order of \(\frac{1}{\sqrt 2 }\left[ {(1 + \sqrt 2 )^{n - 1} - (1 - \sqrt 2 )^{n - 1} } \right]\) $\frac{1}{\sqrt{2}}\left[{(1+\sqrt{2})}^{n-1}-{(1-\sqrt{2})}^{n-1}\right]$ , which exceeds the theoretical limit of \(2^{n - 1}\) $2^{n-1}$ and thereby disproving the almost 50-year-old Kung–Traub conjecture. As the conjecture has always served as a basis for designing optimal algorithms to solve nonlinear equations, this finding is crucial as it implies the potential to devise substantially faster algorithms, which could find practical applications in engineering fields such as CFD, as demonstrated in this paper.