Research On Jacobian-free Newton Krylov Method For High-order CFD Applications And Its Parallel Computing

With the development of Computational Fluid Dynamics(CFD),especially the application of high-order accurate schemes and the increasing in flow mechanism complexity and temporal-spatial definition of flow problems,the computational efficiency requirement of time stepping method becomes higher and higher,and the need for parallel computing is unprecedented.For traditional implicit time stepping methods,the Left-Hand-Side(LHS)approximate Jacobian matrix is usually obtained by low-order schemes,when the Right-Hand-Side(RHS)adopts high-order accurate schemes,the compute stencil of LHS disaccords with that of RHS,and it will degrade the stability and convergence of time stepping method.Jacobian-Free Newton–Krylov(JFNK)method is a combination of inexact Newton method and Krylov subspace method.With the adoption of matrix free technique,JFNK method only calculates the approximate product of Jacobian matrix and vector using finite difference quotient of the nonlinear function,thus successfully avoid the calculation and store of approximate Jacobian matrix.Preconditioner can largely improve the quality of linear system,and can effectively accelerate the convergence rate of JFNK method.LU-SGS method is an efficient preconditioner,it is widely used in many kinds of iteration method such as JFNK,multigrid,etc.However,due to the inherent data dependent feature of LU-SGS method,it's parallel computing on latest multi-core and many-core shared-memory platforms is a tough challenge.This paper is based on the domestic High-Order Simulator for Aerodynamics(HOSTA)program.JFNK time stepping method with LU-SGS preconditioner is designed and realized to improve the computational efficiency effectively.Then realize and optimize large scale parallel computing on the Tianhe-2 supercomputer system.And finally a novel parallel LU-SGS algorithm is proposed to enhance the performance on shared-memory platforms.Three main work of this paper are as follows:(1)A numerical model of JFNK solver for high-order accurate scheme is designed and further implemented in our domestic high-order accurate HOSTA program.We use the steady flow around a circular cylinder as test case to evaluate the stability and convergence rate of JFNK solving method and traditional implicit time stepping method.Test result shows that JFNK solving method is more stable and efficient than traditional LU-SGS method,JFNK method can adopt larger CFL number and save the overall simulation wall time by half.(2)Large scale MPI parallel computing of high-order accurate schemes CFD simulation program with JFNK solving method is realized and optimized on the Tianhe-2 sypercomputer,The scalability test result shows that both JFNK solving method and LU-SGS method have excellent strong scalability,the parallel efficiencies of them remain over 85% when extend to 64 processes(16 nodes,4 processes per node).(3)The inherent data dependent feature of LU-SGS solving method is investigated,two existing OpenMP shared memory parallel LU-SGS strategies of hyper-plane and pipeline are given,and their performance characteristics are tested and analyzed on several multi/many-core platfomrs including Xeon,FT1500 A and MIC.Afterwards,we propose a novel two-layer pipeline parallel method,we evaluate its performance model and compare to actual performance test results.Performance result shows that pipeline method has better computational efficiency than hyper-plane method,and the two-layer pipeline parallel method has higher performance and better scalability than the original pipeline method on MIC.