In this paper, we apply two fully-discrete local discontinuous Galerkin (LDG) methods to the compressible wormhole propagation. We will prove the stability and error estimates of the schemes. Traditional LDG methods use the diffusion term to control of convection term to obtain the stability for some linear equations. However, the variables in wormhole propagation are coupled together and the whole system is highly nonlinear. Therefore, it is extremely difficult to obtain the stability for fully-discrete LDG methods. To fix this gap, we introduce a new auxiliary variable including both the convection and diffusion terms. Moreover, we also construct a special time integration for the porosity, leading to physically relevant numerical approximations and controllable growth rate of the porosity. With a reasonable growth rate, it is possible to handle the time level mismatch in the first-order fully discrete scheme and obtain the stability of the scheme. For the whole system, we will prove that under weak temporal-spatial conditions, the optimal error estimates for the pressure, velocity, porosity and concentration under different norms can be obtained. Numerical experiments are also given to verify the theoretical results.
Keywords: Local discontinuous Galerkin method, implicit-explicit time-marching scheme, stability; error estimate, compressible wormhole propagation
@article{M2AN_2021__55_3_1103_0,
author = {Guo, Hui and Jia, Rui and Tian, Lulu and Yang, Yang},
title = {Stability and error estimates of local discontinuous {Galerkin} method with implicit-explicit time marching for simulating wormhole propagation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1103--1131},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {3},
doi = {10.1051/m2an/2021020},
mrnumber = {4269466},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021020/}
}
TY - JOUR AU - Guo, Hui AU - Jia, Rui AU - Tian, Lulu AU - Yang, Yang TI - Stability and error estimates of local discontinuous Galerkin method with implicit-explicit time marching for simulating wormhole propagation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 1103 EP - 1131 VL - 55 IS - 3 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021020/ DO - 10.1051/m2an/2021020 LA - en ID - M2AN_2021__55_3_1103_0 ER -
%0 Journal Article %A Guo, Hui %A Jia, Rui %A Tian, Lulu %A Yang, Yang %T Stability and error estimates of local discontinuous Galerkin method with implicit-explicit time marching for simulating wormhole propagation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 1103-1131 %V 55 %N 3 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021020/ %R 10.1051/m2an/2021020 %G en %F M2AN_2021__55_3_1103_0
Guo, Hui; Jia, Rui; Tian, Lulu; Yang, Yang. Stability and error estimates of local discontinuous Galerkin method with implicit-explicit time marching for simulating wormhole propagation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 3, pp. 1103-1131. doi: 10.1051/m2an/2021020
[1] , and , A computational Navier-Stokes fluid-dynamics-simulation study of wormhole propagation in carbonate-matrix acidizing and analysis of factors influencing the dissolution process. SPE J. 22 (2017) 187962. | DOI
[2] and , A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131 (1997) 267–279. | MR | Zbl | DOI
[3] , , and , Superconvergence of the local discontinuous Galerkin method for elliptic problems on cartesian grids. SIAM J. Numer. Anal. 39 (2001) 264–285. | MR | Zbl | DOI
[4] , The Finite Element Method for Elliptic Problem. North Holland (1975). | Zbl | MR
[5] and , The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440–2463. | MR | Zbl | DOI
[6] and , Influence of transport and reaction on wormhole formation in porous media. Fluid Mech. Transp. Phenom. 44 (1998) 1933–1949.
[7] , , , , and , On the ability of a Darcy-scale model to capture wormhole formation during the dissolution of a porous medium. J. Fluid Mech. 457 (2002) 213–254. | Zbl | DOI
[8] , Note on the derivative with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20 (1919) 292–296. | MR | JFM | DOI
[9] , and , Local discontinuous Galerkin method for incompressible miscible displacement problem in porous media. J. Sci. Comput. 71 (2017) 615–633. | MR | DOI
[10] , , , and , High-order local discontinuous Galerkin method for simulating wormhole propagation. J. Comput. Appl. Math. 350 (2019) 247–261. | MR | DOI
[11] , and , Mixed finite element-based fully conservative methods for simulating wormhole propagation. Comput. Methods Appl. Mech. Eng. 298 (2016) 279–302. | MR | DOI
[12] and , Characteristic block-centered finite difference method for simulating incompressible wormhole propagation. Comput. Math. App. 73 (2017) 2171–2190. | MR
[13] and , Block-centered finite difference method for simulating compressible wormhole propagation. J. Sci. Comput. 74 (2017) 1115–1145. | MR | DOI
[14] and , A fully conservative finite difference method for simulating Darcy-Forchheimer compressible wormhole propagation. Numer. Algorithms (2018). | MR
[15] , and , Local discontinuous Galerkin method for the Keller-Segel chemotaxis model. J. Sci. Comput. 73 (2017) 943–967. | MR | DOI
[16] , , and , Wormhole propagation behavior under reservoir condition in carbonate acidizing. Transp. Porous Media 96 (2013) 203–220. | DOI
[17] and , 3D Simulation of Carbonate Acidization with HCl: Comparison with Experiments. Society of Petroleum Engineers (2013).
[18] , and , Application of the carman-kozeny correlation to a highporosity and anisotropic consolidated medium: The compressed expanded natural graphite. Transp. Porous Media 43 (2001) 355–376. | DOI
[19] and , Two-scale continuum model for simulation of wormholes in carbonate acidization. AIChE J. 51 (2005) 3231–3248. | DOI
[20] , and , Modeling the acidizing of a carbonate formation. Fluid Dyn. 45 (2010) 779–786. | Zbl | DOI
[21] and , Wormhole formation in dissolving fractures. J. Gophys. Res. 114 (2009) B06203. | DOI
[22] , , and , An -adaptive local discontinuous galerkin method for simulating wormhole propagation with Darcy-Forcheiner model. J. Sci. Comput. 82 (2020) 43. | MR | DOI
[23] , and , Stability and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for advection-diffusion problems. SIAM J. Numer. Anal. 53 (2015) 206–227. | MR | DOI
[24] , and , Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for nonlinear convection-diffusion problems. Appl. Math. Comput. 272 (2016) 237–258. | MR
[25] , , and , Local discontinuous Galerkin methods with implicit-explicit time marching for multi-dimensional convectiondiffusion problems. ESAIM: M2AN 50 (2016) 1083–1105. | MR | Numdam | DOI
[26] , , , and , Local Discontinuous Galerkin method with implicit-explicit time marching for incompressible miscible displacement problem in porous media. J. Sci. Comput. 78 (2018) 1–28. | MR | DOI
[27] , , and , Local discontinuous Galerkin methods with implicit-explicit time-marching for time-dependent incompressible fluid flow. Math. Comput. 88 (2019) 91–121. | MR | DOI
[28] , , and , Local discontinuous Galerkin methods with explicit-implicit-null time discretizations for solving nonlinear diffusion problems. Sci. China (Math.) 063 (2020) 183–204. | MR | DOI
[29] , and , Modeling and analysis on the effect of two-phase flow on wormhole propagation in carbonate acidizing. SPE J. 22 (2017).
[30] , and , Parallel simulation of wormhole propagation with the Darcy–Brinkman–Forchheimer framework. Comput. Geotech. 69 (2015) 564–577. | DOI
[31] and , Local discontinuous Galerkin methods for nonlinear Schrödinger equations. J. Comput. Phys. 205 (2005) 72–97. | MR | Zbl | DOI
[32] and , Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations. Comput. Methods Appl. Mech. Eng. 196 (2007) 3805–3822. | MR | Zbl | DOI
[33] , and , High-Order bound-preserving discontinuous Galerkin methods for wormhole propagation on triangular meshes. J. Comput. Phys. 390 (2019) 323–341. | MR | DOI
[34] and , Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. J. Sci. Comput. 17 (2002) 27–47. | MR | Zbl | DOI
[35] and , A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40 (2002) 769–791. | MR | Zbl | DOI
[36] , , and , Conservative local discontinuous Galerkin method for compressible miscible displacements in porous media. J. Sci. Comput. 73 (2017) 1249–1275. | MR | DOI
[37] , , , and , Characteristic splitting mixed finite element analysis of compressible wormhole propagation. Appl. Numer. Math. 147 (2020) 66–87. | MR | DOI
[38] , , , , and , Theoretical and numerical analyses of chemical-dissolution front instability in fluid-saturated porous rocks. Int. J. Numer. Anal. Methods Geomech. 32 (2008) 1107–1130. | Zbl | DOI
Cité par Sources :
