Numerical passage from kinetic to fluid equations
SIAM Journal on Numerical Analysis
Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions
SIAM Journal on Numerical Analysis
On convergence of numerical schemes for hyperbolic conservation laws with stiff source terms
Mathematics of Computation
Diffusive Relaxation Schemes for Multiscale Discrete-Velocity Kinetic Equations
SIAM Journal on Numerical Analysis
Numerical Schemes for Hyperbolic Systems of Conservation Laws with Stiff Diffusive Relaxation
SIAM Journal on Numerical Analysis
Monotone Difference Approximations Of BV Solutions To Degenerate Convection-Diffusion Equations
SIAM Journal on Numerical Analysis
Discrete Kinetic Schemes for Multidimensional Systems of Conservation Laws
SIAM Journal on Numerical Analysis
A class of approximate Riemann solvers and their relation to relaxation schemes
Journal of Computational Physics
Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems
Mathematics of Computation
Discrete Maximum Principle and Adequate Discretizations of Linear Parabolic Problems
SIAM Journal on Scientific Computing
High-Order Relaxation Schemes for Nonlinear Degenerate Diffusion Problems
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
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Diffusive relaxation systems provide a general framework to approximate nonlinear diffusion problems, also in the degenerate case (Aregba-Driollet et al. in Math. Comput. 73(245):63---94, 2004; Boscarino et al. in Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, 2011; Cavalli et al. in SIAM J. Sci. Comput. 34:A137---A160, 2012; SIAM J. Numer. Anal. 45(5):2098---2119, 2007; Naldi and Pareschi in SIAM J. Numer. Anal. 37:1246---1270, 2000; Naldi et al. in Surveys Math. Indust. 10(4):315---343, 2002). Their discretization is usually obtained by explicit schemes in time coupled with a suitable method in space, which inherits the standard stability parabolic constraint. In this paper we combine the effectiveness of the relaxation systems with the computational efficiency and robustness of the implicit approximations, avoiding the need to resolve nonlinear problems and avoiding stability constraints on time step. In particular we consider an implicit scheme for the whole relaxation system except for the nonlinear source term, which is treated though a suitable linearization technique. We give some theoretical stability results in a particular case of linearization and we provide insight on the general case. Several numerical simulations confirm the theoretical results and give evidence of the stability and convergence also in the case of nonlinear degenerate diffusion.