High Order Finite Difference Multi-resolution WENO Method for Nonlinear Degenerate Parabolic Equations

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

“…We show the evolution of the numerical solution at different time in Figure 2. From the results, we can observe that the numerical solutions don't appear noticeable oscillation around the interface and agree very well with the reference solution in [24,27].…”

“…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).…”

Sixth order weighted essentially non-oscillatory schemes with Z-type nonlinear weighting procedure for nonlinear degenerate parabolic equations

“…Further, the hybrid scheme based on the spatial MWENO and the temporal NRK schemes was employed to solve the equation (1.1) numerically. Recently, Rathan et al [17] proposed a new type of local and global smoothness indicators in L 1 norm based on the concept of achieving a higher order approximation to the lower order derivatives through undivided differences and subsequently constructed the new Z-type nonlinear weights, and Jiang [12] designed an alternative formulation to approximate the second derivatives in a conservative form, where the odd order derivatives at half points were used to construct the numerical flux.…”

High order finite difference WENO methods with unequal-sized sub-stencils for the Degasperis-Procesi type equations

“…The WENO schemes with unequal-sized sub-stencils are particularly attractive because of their simplicity both in the choice of the stencil and in the freedom of arbitrary positive linear weights, especially for unstructured meshes. We refer reader to [1,19] for WENO schemes with unequal-sized sub-stencils for solving degenerate parabolic equations which involves second order derivatives. In this paper, we generalize WENO schemes with unequal-sized sub-stencils to the DP type equations, which involve nonlinear high order derivatives (> 2).…”

“…More discussions regarding this recipe can be found in [2,5,12,47]. 19) is computed using p1 (x) defined in (2.15), while β 1 is computed using p 1 (x) defined in (2.11) in [47]. Both choices obtain high-order accuracy and equally good nonoscillatory results for all of our numerical simulations except the µDP equation with shock solutions, in which the numerical solution obtained with β 1 computed by p1 (x) is essentially non-oscillatory while the numerical solution obtained with β 1 computed by p 1 (x) has over-and under-shoots.…”