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In this paper, we introduce Super Luminal ("xLumi"), a new payment channel protocol for blockchain systems. xLumi is a simple unidirectional payment channel that can be extended to a bidirectional payment channel or a complete network.xLumi guarantees the security of the payment channel's funds by using a simple set of mathematical rules that can be easily implemented on any blockchain with the necessary infrastructure. We also give the detailed implementation methods of this idea using V Systems contract systems in this paper.

We develop an effective interface tracking method to simulate the incompressible Stokes flow with moving interfaces. The Stokes equations are first rewritten into a system of elliptic equations with singular sources which can be efficiently solved by a simple weak formulation proposed in [11]. The key idea is to first split the solution into a singular part and a regular part additively. The singular part captures the interface conditions, while the regular part approximates the equations in the whole domain, which can be solved by the standard finite element formulation. We carefully design numerical methods to interpolate the velocity to the moving interface. Numerical tests are carried out to demonstrate the accuracy and other properties of our method.

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