Hyperbolic conservation laws are often used as models in science and engineering. The application might be gas dynamics, acoustics, porous media flow or other systems involving wave propagation. Examples of hyperbolic conservation laws include the Euler equations for ideal gas dynamics, the shallow water equations, the equations of magnetohydrodynamics (MHD), the advection equation, the Buckley-Leverett equation for porous media flow, and many more. There are special properties and mathematical difficulties associated with these equations that are not seen elsewhere, for instance, the formation of discontinuous solutions (shock waves, contact discontinuities, slip lines, etc) and nonuniqueness of solutions. These difficulties must be dealt with carefully whenever numerical methods are developed. On the other hand, the equations have a rich mathematical structure that can be exploited to develop efficient methods. (A typical numerical approach involves using the property of local conservation).

In the last decades, a large number of high-resolution methods have been developed to capture the discontinuous solutions that are typical for hyperbolic conservation laws. Recently, there has been an increasing focus on genuinely multidimensional methods that are able to represent the geometrically complex interaction of linear and nonlinear waves. (This complexity can be seen for fairly simple problems, as for instance the two-dimensional Riemann problem for gas dynamics, or a strong shock hitting a low-density bubble.)

We are active participants within the conservation laws community and have close contacts with leading experts. We can therefore offer expertise on mathematical properties for conservation laws and on modern high-resolution methods (Godunov, TVD, (W)ENO, wave propagation, nonoscillatory central difference, etc.). In particular, we offer world-leading expertise on front-tracking and operator splitting methods, central difference methods, and applications within porous media flow.

Joint with NTNU we run a preprint archive, the Conservation Laws Preprint Server, which gives a broad overview of current activity within this field. More details can also be found on the pages our BeMatA project: Nonlinear partial differential equations of evolution type - theory and numerics.

Selected publications:

• H. Holden, K.-A. Lie, and N. H. Risebro. An unconditionally stable method for the Euler equations. . Comput. Phys. Vol. 150, no. 1, pp. 76-96, 1999. (additional pictures and animations)
• H. Holden, K. Hvistendahl Karlsen, and K.-A. Lie. Operator splitting methods for degenerate convection-diffusion equations II: numerical examples with emphasis on reservoir simulation.Computational Geosciences, Vol. 4, No. 4, pp. 287-322, 2000.
• K.-A. Lie and S. Noelle. An improved quadrature rule for the flux-computation in staggered central difference schemes in multidimensions.J. Sci. Comp., Vol. 18, No. 1, pp. 69-81, 2003.
• K.-A. Lie and S. Noelle. On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws.SIAM J. Sci. Comp., Vol. 24, No. 4, pp. 1157-1174, 2003.

"Abuse" of the new SINTEF logo: the logo filled with low density gas is deformed by a Mach 3 shock. Computation using a high-resolution central difference scheme, see Lie & Noelle.

A dambreak problem for the the shallow water equations. From Holdahl, Holden, and Lie

An "explosion" in ideal gas dynamics: a cylinder of high pressure within two parallel walls computed using a front-tracking method.