  We present a general, high-order, fully explicit relaxation scheme which can
be applied to any system of nonlinear hyperbolic conservation laws in multiple
dimensions. The scheme consists of two steps. In a first (relaxation) step, the
nonlinear hyperbolic conservation law is approximated by a kinetic equation
with stiff BGK source term. Then, this kinetic equation is integrated in time
using a projective integration method. After taking a few small (inner) steps
with a simple, explicit method (such as direct forward Euler) to damp out the
stiff components of the solution, the time derivative is estimated and used in
an (outer) Runge-Kutta method of arbitrary order. We show that, with an
appropriate choice of inner step size, the time step restriction on the outer
time step is similar to the CFL condition for the hyperbolic conservation law.
Moreover, the number of inner time steps is also independent of the stiffness
of the BGK source term. We discuss stability and consistency, and illustrate
with numerical results (linear advection, Burgers' equation and the shallow
water and Euler equations) in one and two spatial dimensions.
