Finite volume WENO schemes for nonlinear parabolic problems with degenerate diffusion on non-uniform meshes

An Eulerian-Lagrangian Runge-Kutta Finite Volume (El-Rk-Fv) Method for Solving Convection and Convection-Diffusion Equations

“…The authors in [5] write WENO-AO(p, r) to denote an adaptive order that is at best pth order (from the large center stencil) and at worst rth order (from the stable lower order stencils). For our purposes we use WENO-AO (5,3). The end product of WENO and WENO-AO methods is ultimately a reconstruction polynomial R j (x ∈ I j ) that we shall use for reconstruction.…”

“…Mass conservation is also numerically verified by applying the proposed algorithm to the 0D2V (zero dimensions in physical space and two dimensions in velocity space) Leonard-Bernstein Fokker-Planck equation. We assume a uniform mesh, apply WENO-AO (5,3) in Algorithms 1 and 2, use three Gauss-Legendre nodes in Algorithm 3, and use the fourth-order approximation given by equation (3.4) for the diffusive term. Unless otherwise stated, for the time-stepping we use the fourth-order RK method for pure convection problems, and IMEX(2,3,3) for convection-diffusion equations.…”

“…The recently developed WENO-AO schemes [4,5,6] are robust and guarantee the existence of the linear weights at arbitrary points. We note that Chen, et al used WENO-AO schemes in the SL framework [15], and Huang and Arbogast have recently used WENO-AO schemes in the Eulerian framework [3,4]. RK methods are used to evolve the MOL system along the approximate characteristics.…”

A high‐order weighted essentially nonoscillatory scheme based on exponential polynomials for nonlinear degenerate parabolic equations

“…The existence and uniqueness of the weak answers for this group of equations have been discussed in [6,11,38,44]. In recent decades, finding numerical solutions of Equation (1.1) using different ideas has attracted the attention of researchers such as relaxation scheme [18], adaptive multiresolution schemes [12,15,16], finite volume schemes [8,13,19,35], kinetic scheme [9], local discontinuous Galerkin finite element method [46], mixed finite element methods [7,37,39] and WENO schemes [3,25,33,40]. In recent decades, numerical solution of partial differential equations has been considered by many researchers [1,2,4,5,29,30,34,43].…”

An oscillation free local discontinuous Galerkin method for nonlinear degenerate parabolic equations

“…In fact, the spurious oscillations may occur near the interfaces and wave fronts that are harmful to the robustness of the numerical algorithm. To overcome this difficulty, various schemes and approaches have been developed in the literature, such as interface tracking algorithms [20], diffusive kinetic schemes [3], relaxation schemes [10], finite difference/volume weighted essentially nonoscillatory (WENO) methods [2,24,27], entropy stable schemes with artificial viscosity [23], method of lines transpose (MOL T ) approach with nonlinear filters [12], discontinuous Galerkin (DG) methods with maximum-principle-satisfying limiters [36,41], local DG finite element methods [40], direct DG methods [30], etc. In this paper, we focus on the DG method and extend our previous work [31] to the nonlinear convection-diffusion problem (1.1).…”