A Discrete Kinetic Approximation of Entropy Solutions to Multidimensional Scalar Conservation Laws

Abstract

We present a new relaxation approximation to scalar conservation laws in several space variables by means of semilinear hyperbolic systems of equations with a finite number of velocities. Under a suitable multidimensional generalization of the Whitham relaxation subcharacteristic condition, we show the convergence of the approximated solutions to the unique entropy solution of the equilibrium Cauchy problem. Academic Press

On the Transverse Diffusion of Beams of Photons in Radiation Therapy

“…In practice, the larger is the λ i the more stable is the resulting scheme, but the more diffusive it is. This can be verified by reproducing the computations of [14,1]. The present correction uses similar ideas as the ones proposed in [6,2].…”

“…The computational time is although higher when using the new relaxation parameters (15) than when using the one in (11). This is actually due to the method used to compute Ψ n l,m from (14) or (16). The conditioning of problem (14) is simply better than the one of (16) which explains the difference of computational times.…”

“…In order to handle the non-linearity in (8), the relaxation approach proposed in [16] and based on the previous work of [14,1] is used.…”

“…in the scheme (14). The numerical fluxes are now defined locally as a function of the unknowns and the fluxes which allows to better capture the diffusion effects.…”

Numerical Schemes of Diffusion Asymptotics and Moment Closures for Kinetic Equations

“…In order to prevent an initial layer from appearing [42] in the α−splitting, we need to prescribe well-prepared initial conditions taking into account both splittings as:…”