摘要
JFNK(Jacobian-free Newton-Krylov)方法是由Newton迭代方法和Krylov子空间迭代方法构成的嵌套迭代方法。作者以全隐式差分的一维非线性平流方程(亦称无粘Burgers方程)探讨采用全隐式格式计算的必要性和JFNK方法的有效性。模拟结果表明,隐式结果比显式和半隐式结果在稳定度和精度方面较大的优越性,特别是模拟气流强的系统以及要素空间分布具有较大梯度的系统。
Over the last 10 years it had major advantages in super computing that the higher and higher resolution was adopted in atmospheric numerical models for various scales. And this in turn provides possibility and necessity for using an implicit schema Zeng and Ji proposed a scheme with energy conservation in 1981. However, the convergence of iteration involved in the fully implicit scheme remains a key issue to solve. JFNK （Jacobian-free NewtonKrylov） method, a highly efficient method for solving implicit nonlinear equation, has been successfully used in many fields of computational physics.
JFNK method is a nested iteration method for solving the nonlinear partial differential equations （PDEs）, which is synergistic combination of Newton-type method and Krylov subspace method, without forming and storing the elements of the true Jacobian matrix. Some studies have shown that the fully implicit method solved by JFNK is more efficient and accurate than the semi-implicit method for a given level of accuracy and for a given CPU time.
This paper, based on the 1D nonlinear advection equation （namely the inviscid Burgers equation）, a number of comparative simulations, with the 2-order fully implicit （IM） schemes, the 2-order explicit （EX） schemes and 2-order semi-implicit （SI） schemes, have been carried out to investigate the necessity of adopting fully implicit schemes and the effect of JFNK method. The results show that the implicit scheme has better performance in comparison with the other two schemes in terms of computational stability and accuracy, especially for the simulation of flow system with strong current or large gradient of variables. These can be summarized as follows.
（1） When the time step is smaller （say time step is 2^-9）, all the three schemes are stable; but the accuracy of IM scheme is higher than it of the others. And with the increase of time step, the IM scheme remains stable, while the other two become unstable up to time step is 2^-3 when basic flow is 0. 0,
出处
《大气科学》
CSCD
北大核心
2007年第5期963-972,共10页
Chinese Journal of Atmospheric Sciences
基金
国家自然科学基金资助项目40545020
