In the simulation of complex flows with large-scale or long-time integration, it is
required that the numerical method is of high-order accuracy and high resolution, and also the
computing process is always strong stable and high efficient. That means we should develop
the high efficiency and high accuracy numerical methods for both the spatial and temporal
discretizations for this kind of simulation. In recent years, the weighted essentially
non-oscillatory(WENO) scheme is widely applied in CFD and continuously developed by
researchers due to its excellent high-accuracy high-resolution ability for capturing complex
flow characteristics. For the temporal integration, the Runge-Kutta method, which is well
known as a classical method for ordinary differential equation, is usually used in CFD due to
its high-accuracy, good-stability and low-storage. In this paper, we propose several methods
to improve the performance of the WENO schemes and the Runge-Kutta method.
1)A shared-weights WENO method for solving the Euler equations is proposed. The
shared weights are calculated by using the energy flux components of Steger-Warming split
method. Since the weights are shared by all flux components, the total computing efficiency
is greatly improved. The variable used to calculate the shared weights plays an important role
in the property of the WENO scheme, an inappropriate selection may result in an unstable
process or low-resolution results. Numerical experiments showed that the shared-weights
WENO method proposed in this paper has good robustness and low dissipation error.
2)An improved multi-step CRWENO scheme is proposed. Multi-step CRWENO scheme
has spectral-like resolution as compact schemes in smooth regions, and meanwhile can obtain
high accuracy and resolution around discontinuities. However, it has the same drawback of
expensive computing cost as original multi-step WENO scheme. In this paper we extend the
similar idea recently proposed by Zeng et al for improving the multi-step WENO scheme
to the multi-step CRWENO scheme, and an improved multi-step CRWENO scheme is
constructed. Numerical results of different schemes are compared and the performance of this
new scheme is verified numerically.
3)This paper has developed a Runge-Kutta scheme with a last implicit stage. The
Runge-Kutta method has advantages such as high accuracy, good stability and low storage.
However, the time step of the explicit Runge-Kutta scheme is limited by the stability
requirement. In order to increase the time step and stability, we have proposed a kind of
Runge-Kutta method with a last implicit stage. Compared to explicit RK scheme, the new
method can take a larger time step for calculation while keeping stable and low dissipation
error, this is helpful to improve the efficiency when it comes to simulation of stiff problems.
In addition, only one implicit stage means its computation cost is apparently less than the
diagonally implicit Runge-Kutta scheme. Using the analysis of stability and resolution
property to determine coefficients, we have developed the third-order and fourth-order
Runge-Kutta schemes with a last implicit stage. With a spatial shock-capturing scheme, the new RK schemes are applied in different hyperbolic equations. Numerical results have
proved our theoretical analyses.