Total-variation-diminishing time discretizations
SIAM Journal on Scientific and Statistical Computing
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
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations
Applied Numerical Mathematics - Special issue on time integration
Relaxation Schemes for Nonlinear Kinetic Equations
SIAM Journal on Numerical Analysis
Total variation diminishing Runge-Kutta schemes
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
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
A class of approximate Riemann solvers and their relation to relaxation schemes
Journal of Computational Physics
Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems
SIAM Journal on Numerical Analysis
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
SIAM Journal on Numerical Analysis
An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems
SIAM Journal on Numerical Analysis
Thehp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations
Mathematics of Computation
Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems
Mathematics of Computation
Implicit---Explicit Runge---Kutta Schemes and Applications to Hyperbolic Systems with Relaxation
Journal of Scientific Computing
Journal of Scientific Computing
High-Order Relaxation Schemes for Nonlinear Degenerate Diffusion Problems
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Linearly Implicit Approximations of Diffusive Relaxation Systems
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
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In this work we present finite element approximations of relaxed systems for nonlinear diffusion problems, which can also tackle the cases of degenerate and strongly degenerate diffusion equations. Relaxation schemes take advantage of the replacement of the original partial differential equation (PDE) with a semilinear hyperbolic system of equations, with a stiff source term, tuned by a relaxation parameter $\epsilon$. When $\epsilon \rightarrow 0^+$, the system relaxes onto the original PDE: in this way, a consistent discretization of the relaxation system for vanishing $\epsilon$ yields a consistent discretization of the original PDE. The numerical schemes obtained with this procedure do not require solving implicit nonlinear problems and possess the robustness of upwind discretizations. The proposed approximations are based on a discontinuous Galerkin method in space and on suitable implicit-explicit integration in time. Then, in principle, we can achieve any order of accuracy and obtain stable solutions, even when the diffusion equation becomes degenerate and solution singularities develop. Moreover, when needed, we can easily incorporate slope limiters within our schemes in order to handle spurious oscillatory phenomena. Some preliminary theoretical results are given, along with several numerical tests in one and two space dimensions, both for linear and nonlinear diffusion problems, including a degenerate diffusion equation, that provide numerical evidence of the properties of the presented approach.