On a smooth volume approach, integral conservation law, and upwind scheme with monotonic reconstruction

This paper presents a smooth volume approach – an alternative to smooth particle formalism. The proposed approach is based on approximation of integral conservation law and can be viewed as a generalization for the finite volume method. To provide insight into properties of smooth volume schemes, a hypothesis is presented. On its basis, an extension technique for the development of smooth volume schemes is suggested. Using this technique, the finite volume upwind and the Godunov schemes with monotonic reconstruction are generalized to smooth volume schemes. The hypothesis, the technique and the resulting schemes are tested by applying them to gasdynamics shock tube problems. Precise and monotonic calculation results verify validity of our theory and properties of the schemes developed.