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

A high-order relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws