  The weighted essentially non-oscillatory (WENO) schemes are a popular class
of high order accurate numerical methods for solving hyperbolic partial
differential equations (PDEs). The computational cost of such schemes increases
significantly when the spatial dimensions of the PDEs are high, due to large
number of spatial grid points and nonlinearity of high order accuracy WENO
schemes. How to achieve fast computations by WENO methods for high spatial
dimension PDEs is a challenging and important question. Recently, sparse-grid
has become a major approximation tool for high dimensional problems. The open
question is how to design WENO computations on sparse grids such that
comparable high order accuracy of WENO schemes in smooth regions and
essentially non-oscillatory stability in non-smooth regions of the solutions
can still be achieved as that for computations on regular single grids? In this
paper, we combine the third order finite difference WENO method with
sparse-grid combination technique and solve high spatial dimension hyperbolic
equations on sparse grids. WENO interpolation is proposed for the prolongation
part in sparse grid combination techniques to deal with discontinuous solutions
of hyperbolic equations. Numerical examples are presented to show that
significant computational times are saved while both high order accuracy and
stability of the WENO scheme are maintained for simulations on sparse grids.
