A new category of Hermitian upwind schemes for computational acoustics

A new high-order upwind scheme is developed to solve linear acoustics. In order to produce a compact scheme with a high level of precision, a Hermite interpolation is used. An algorithm in three stages is built in order to optimize the procedure of discretization. In the “reconstruction stage”, a local spatial interpolator based on an upwind stencil is defined. In the “decomposition stage”, the calculation of a wave model adapted to the decomposition of the time-derivatives in simple wave contributions is derived; then the local spatial interpolator is used in order to calculate the wave strengths for each wave. In the “evolution stage”, we update by means of a procedure similar to that of Cauchy–Kovaleskaya, the discrete variable and its derivatives, by using the information brought by each simple wave. Specific boundary conditions are finally established. In this way, a third-order upwind Hermitian scheme may be constructed (“Δ-P3 scheme”). A one-dimensional analysis in Fourier series shows the principal properties of such a scheme. The effectiveness and the exactitude of the new compact scheme are shown by their applications to several two-dimensional problems and by comparisons to competing finite difference method for wave propagation.