scholarly journals Stability-enhanced AP IMEX1-LDG Method: Energy-based Stability and Rigorous AP Property

2021 ◽  
Vol 59 (2) ◽  
pp. 925-954
Author(s):  
Zhichao Peng ◽  
Yingda Cheng ◽  
Jing-Mei Qiu ◽  
Fengyan Li
Keyword(s):  
Ldg Method

LDG method for reaction–diffusion dynamical systems with time delay

2011 ◽  
Vol 217 (22) ◽  
pp. 9173-9181 ◽  
Author(s):  
Dongfang Li ◽  
Chengjian Zhang ◽  
Hongyu Qin
Keyword(s):  
Time Delay ◽  
Dynamical Systems ◽  
Reaction Diffusion ◽  
Ldg Method

Evaluating the interaction of biofilms, organic matter and soil structures at the pore scale

2021 ◽  
Author(s):  
Alexander Prechtel ◽  
Simon Zech ◽  
Alice Lieu ◽  
Raphael Schulz ◽  
Nadja Ray
Keyword(s):  
Organic Matter ◽  
Structural Changes ◽  
Effective Diffusivity ◽  
Computational Domain ◽  
Pore Scale ◽  
Gas Phases ◽  
Soil Structures ◽  
Ldg Method ◽  
Conventional Models ◽  
Diffusive Processes

<div class="description js-mathjax"> <p>Key functions of soils, such as permeability or habitat for microorganisms, are determined by structures at the microaggregate scale. The evolution of elemental distributions and dynamic processes can often not be assessed experimentally. So mechanistic models operating at the pore scale are needed.<br />We consider the complex coupling of biological, chemical, and physical processes in a hybrid discrete-continuum modeling approach. It integrates dynamic wetting (liquid) and non-wetting (gas) phases including biofilms, diffusive processes for solutes, mobile bacteria transforming into immobile biomass, and ions which are prescribed by means of partial differential equations. Furthermore the growth of biofilms as, e.g., mucilage exuded by roots, or the distribution of particulate organic matter in the system, is incorporated in a cellular automaton framework (CAM) presented in [1, 2]. It also allows for structural changes of the porous medium itself (see, e.g. [3]). As the evolving computational domain leads to discrete discontinuities, we apply the local discontinuous Galerkin (LDG) method for the transport part. Mathematical upscaling techniques incorporate the information from the pore to the macroscale [1,4].<br />The model is applied for two research questions: We model the incorporation and turnover of particulate OM influencing soil aggregation, including ‘gluing’ hotspots, and show scenarios varying of OM input, turnover, or particle size distribution. <br />Second, we quantify the effective diffusivity on 3D geometries from CT scans of a loamy and a sandy soil. Conventional models cannot account for natural pore geometries and varying phase properties. Upscaling allows also to quantify how root exudates (mucilage) can significantly alter the macroscopic soil hydraulic properties.</p> </div> <div id="field-23"> <p>[1]  Ray, Rupp, Prechtel (2017). AWR (107), 393-404.<br />[2] Rupp, Totsche, Prechtel, Ray (2018). Front. Env. Sci. (6) 96.<br />[3] Zech, Dultz, Guggenberger, Prechtel, Ray (2020). Appl. Clay Sci. 198, 105845.<br />[4] Ray, Rupp, Schulz, Knabner (2018). TPM 124(3), 803-824.</p> </div>

scholarly journals An improved superconvergence error estimate for the LDG method

2015 ◽  
Vol 98 ◽  
pp. 122-136
Author(s):  
Slimane Adjerid ◽  
Nabil Chaabane
Keyword(s):  
Error Estimate ◽  
Ldg Method

Local Discontinuous Galerkin Methods for the Degasperis-Procesi Equation

2011 ◽  
Vol 10 (2) ◽  
pp. 474-508 ◽  
Author(s):  
Yan Xu ◽  
Chi-Wang Shu
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Galerkin Methods ◽  
Sharp Transition ◽  
Piecewise Constant ◽  
Local Discontinuous Galerkin ◽  
Local Discontinuous Galerkin Methods ◽  
Different Types ◽  
Ldg Method ◽  
Nonlinear High Order

AbstractIn this paper, we develop, analyze and test local discontinuous Galerkin (LDG) methods for solving the Degasperis-Procesi equation which contains nonlinear high order derivatives, and possibly discontinuous or sharp transition solutions. The LDG method has the flexibility for arbitrary h and p adaptivity. We prove the L2 stability for general solutions. The proof of the total variation stability of the schemes for the piecewise constant P0 case is also given. The numerical simulation results for different types of solutions of the nonlinear Degasperis-Procesi equation are provided to illustrate the accuracy and capability of the LDG method.

scholarly journals Convergence Analysis of the LDG Method for Singularly Perturbed Reaction-Diffusion Problems

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2291
Author(s):  
Yanjie Mei ◽  
Sulei Wang ◽  
Zhijie Xu ◽  
Chuanjing Song ◽  
Yao Cheng
Keyword(s):  
Layer Structure ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
Energy Norm ◽  
Approximation Properties ◽  
Anisotropic Meshes ◽  
Balanced Norm ◽  
Ldg Method ◽  
Local Projections ◽  
Diffusion Problems

We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction–diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and “balanced” norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the “balanced” norm is artificially introduced to capture the boundary layer contribution. We establish a uniform convergence of order k for the LDG method using the balanced norm with the local weighted L2 projection as well as an optimal convergence of order k+1 for the energy norm using the local Gauss–Radau projections. The numerical method, the layer structure as well as the used adaptive meshes are all discussed in a symmetry way. Numerical experiments are presented.

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