Local discontinuous Galerkin methods with explicit Runge‐Kutta time marching for nonlinear carburizing model

Abstract

A fully discrete local discontinuous Galerkin (LDG) method coupled with 3 total variation diminishing Runge-Kutta time-marching schemes, for solving a nonlinear carburizing model, will be analyzed and implemented in this paper. On the basis of a suitable numerical flux setting in the LDG method, we obtain the optimal error estimate for the Runge-Kutta-LDG schemes by energy analysis, under the condition τ ≤ λh 2 , where h and τ are mesh size and time step, respectively, λ is a positive constant independent of h. Numerical experiments are presented to verify the accuracy and capability of the proposed schemes. For the carburizing diffusion processes of steel and the diffusion simulation for Cu-Ni system, the numerical results show good agreement with the experimental results.

L1/LDG algorithm for time‐Caputo space‐Riesz fractional convection equation in two dimensions

“…). Let 𝚷 be the projection defined by Equations ( 21) and (22). Then for any 𝛒 ∈ H r (Ω) with 1 ≤ r ≤ k+1, there holds…”

“…Here the orthogonal properties of the projections are applied. The choice of numerical fluxes ( 10) and ( 11), Lemma 6, and Equations ( 19) and (22)…”

“…The discontinuous Galerkin (DG) method was originally proposed to solve hyperbolic conservation laws containing only first‐order spatial derivatives [18] and was latter generalized to the local discontinuous Galerkin (LDG) method by Cockburn and Shu to solve differential equations with higher‐order derivatives [19, 20]. Nowadays, DG method and LDG method are recognized as well‐established methods for classical conservation laws [21] and become adequate instruments for solving differential equations [22]. In particular, the LDG method is widely adopted due to the geometric flexibility, the support of locally adapted resolution, as well as excellent parallel efficiency.…”