High Order WENO Finite Volume Schemes Using Polyharmonic Spline Reconstruction

Polyharmonic splines are utilized in the WENO reconstruction of finite volume discretizations, yielding a numerical method for scalar conservation laws of arbitrary high order. The resulting WENO reconstruction method is, unlike previous WENO schemes using polynomial reconstructions, numerically stable and very flexible. Moreover, due to the theory of polyharmonic splines, optimal reconstructions are obtained in associated native Sobolev-type spaces, called Beppo Levi spaces. This in turn yields a very natural choice for the oscillation indicator, as required in the WENO reconstruction method. The key ingredients of the proposed polyharmonic splineWENO reconstruction algorithm are explained in detail, and one numerical example is given for illustration.