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This paper focuses on the study of the finite-time consensus (FTC) and fixed-time consensus (FXC) issues of multi-agent systems (MASs) driven by parabolic partial differential equations (PDEs). Compared with the study in the existing literature, the topic of FTC and FXC control is first embodied in MASs driven by parabolic PDEs. Based on the Lyapunov theorems, the FTC and FXC controllers are devised to ensure that the MASs converge to a stable state with external disturbance. Furthermore, we simplify the controllers to guarantee the FTC and FXC of MASs without external disturbance. Finally, two illustrative examples are given to verify the feasibility of controllers.

In this paper, we introduce and study a stochastic differential variational inequality (SDVI) which consists of a stochastic differential equation and a stochastic variational inequality. We obtain the existence and uniqueness of the solutions for SDVI by using the iteration method and Gronwall's inequality. Moreover, we show the convergence of Euler scheme for solving SDVI under some mild conditions. Finally, we apply the obtained results to solve the electrical circuits with diodes and the collapse of the bridge problems in stochastic environment.

Roe scheme is known for its good performance in moderate-Mach-number flows. However, this scheme and its extended versions suffers from many disastrous problems, such as non-physical behavior, global cut-off, and checkerboard problems, for incompressible flows; and shock instability, expansion shock, and positively non-conservative problems for hypersonic flows. In this paper, non-physical behavior problem, checkerboard problem, and main reason of shock instability problem are due to that the Roe scheme cannot identify multi-dimensional incompressible and compressible flows when normal Mach number on the cell face tends to zero, and then leads to incorrect cross modifications. Positively non-conservative problem is also identified as another important reason for shock instability. Therefore, Mach number and an assistant pressure-density-varying detector are introduced into the Roe scheme to judge compressibility, positivity condition is satisfied by a simple modification with minimal numerical dissipation increases and even with possible decreases in numerical dissipation, the mechanism of the preconditioned Roe scheme is introduced to suppress checkerboard problem, and modified entropy fix and the rotated Riemann solver is combined with complementary advantages as an assistant improvement for better robust. Based on above improvements and previous developments for global cut-off and expansion shock problems, an improvement Roe scheme for all Mach-number flow (Roe-AM) is proposed to simultaneously overcome nearly all well-known drawbacks of the classical Roe scheme. The Roe-AM scheme is simple, easy to implement, computationally low-cost, robust, good extensibility, and free of empirical parameters essentially, with increasing minimal numerical dissipation.

A common defect of the Roe scheme is the production of non-physical expansion shock and shock instability. An improved method with several advantages was presented to suppress the shock instability. However, this method cannot prevent expansion shock and is incompatible with the traditional curing method for expansion shock. Therefore, the traditional curing mechanism is analyzed. The discussion explains the effectiveness of the traditional curing method and identifies several defects, one of which leads to incompatibility between curing the shock instability and expansion shock. Consequently, a new improved Roe scheme is proposed in this study. This scheme is concise, easy to implement, low computational cost, and robust. More importantly, the scheme can simultaneously cure the shock instability and expansion shock without additional costs.

The shock instability phenomenon is a famous problem for the shock-capturing scheme. By subdividing the numerical dissipation of the Roe scheme, the term of pressure-difference-driven modification for the cell face velocity is regarded as a version of the momentum interpolation method (MIM), which is necessary for incompressible flows to suppress the pressure checkerboard problem. Through the analysis and odd-even decoupling test, it is discovered that MIM plays the most important role on the shock instability. In fact, for non-linear flows MIM should be completely removed, but unexpected MIM is activated on the cell face nearly parallel to the flow for high Mach number flows or low Mach number flows in shock. For such conditions, two coefficients are designed based on local Mach number and a shock detector, respectively, and then the improved Roe scheme is proposed, which gives consideration to requirement of MIM for incompressible and compressible flows and is validated for good performance of avoiding odd-even decoupling. Therefore, the aim of decreasing rather than increasing numerical dissipation to cure the shock instability can be achieved.

As the representative of the shock-capturing scheme, the Roe scheme fails to LES because important turbulent characteristics cannot be reproduced such as the famous k-5/3 spectral law owing to large numerical dissipation. In this paper, the Roe scheme is divided into five parts: , , , , and , which means basic upwind dissipation, pressure-difference-driven and velocity-difference-driven modification of the interface fluxes and pressure, respectively. Then, the role of each part on LES is investigated by homogeneous decaying turbulence. The results show that the parts , , and have little effect on LES. It is important especially for because it is necessary for computation stability. The large numerical dissipation is due to and , and each of them has much larger dissipation than SGS dissipation. According to these understanding, an improved all-speed LES-Roe scheme is proposed, which can give enough good LES results for even coarse grid resolution with usually adopted reconstruction.

There are many ideas for developing shock-capturing schemes and their extension for all-speed flow. The representatives of them are Roe, HLL and AUSM families. In this paper, a uniform algorithm is proposed, which expresses three families in the same framework. The algorithm has explicit physical meaning, provides a new angel of understanding and comparing the mechanism of schemes, and may play a great role in the further research. As an example of applying the uniform algorithm, the low-Mach number behaviour of the schemes is analyzed. Then, a very clear and simple explanation is given based on the wall boundary, and a concise rule is proposed to judge whether a scheme has satisfied low-Mach number behaviour.