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A method for solving nonlinear differential equations, which facilitates the computation of solutions of a high polynomial degree on a grid, is tested for use in direct numerical simulation (DNS) of two-phase unsteady flow. The method uses a grid discretization to approximate continuously distributed variables, represented by functions of time and space, in a given set of differential equations. The grid contains information about both the values and the values of the derivatives of the unknown functions at the grid points in the computational domain. With this method the derivatives are thus explicitly defined at each grid point rather than, as in conventional numerical schemes, implicitly given by the function values at the surrounding grid points. Using piecewise polynomial interpolation functions can be represented with an arbitrary order of continuity over the entire computational domain. The high polynomial order used in this method allows for simulation of flow features smaller than the interval separating each grid point. This reduces the required number of grid points and the need to adapt the grid to complex boundary geometry or to the interphase between different fluid phases. This simplifies grid generation and reduces the computational cost.

A high-order approach to solving nonlinear differential equations applied to direct numerical simulation of two-phase unsteady flow