A high-performance finite-volume algorithm for solving partial differential equations governing compressible viscous flows on structured grids

This work focuses on the development of a high-performance fourth-order finite-volume method to solve the nonlinear partial differential equations governing the compressible Navier-Stokes equations on a Cartesian grid with adaptive mesh refinement. The novelty of the present study is to introduce the loop chaining concept to this complex fourth-order fluid dynamics algorithm for significant improvement in code performance on parallel machines. Specific operations involved in the algorithm include the finite-volume formulation of fourth-order spatial discretization stencils and optimal inter-loop parallelization strategies. Numerical fluxes of the Navier-Stokes equations comprise the hyperbolic (inviscid) and elliptic (viscous) ​components. The hyperbolic flux is evaluated using high-resolution Godunov's method and the elliptic flux is based on fourth-order centered-difference methods everywhere in the computational domain. The use of centered-difference methods everywhere supports the idea of fusing modular codes to achieve high efficiency on modern computers. Temporal discretization is performed using the standard fourth-order ​Runge-Kutta method. The fourth-order accuracy of solution in space and time is verified with a transient Couette flow problem. The algorithm is applied to solve the Sod's shock tube and the transient flat-plate boundary layer flow. The numerical predictions are validated by comparing to the analytical solutions. The performance of the baseline code is compared to that of the fused scheme which fuses modular codes via loop chaining concept and a significant improvement in execution time is observed.