Stability analysis and error estimates of fully-discrete local discontinuous Galerkin methods for simulating wormhole propagation with Darcy–Forchheimer model

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Department of Mathematical Sciences


In this paper, we apply local discontinuous Galerkin (LDG) methods to compressible wormhole propagation with Darcy–Forchheimer model. We consider two time integrations up to second-order accuracy and prove the stability of the fully-discrete schemes. There are several difficulties. Firstly, different from most previous works discussing stability of wormhole propagations, we use LDG methods and have to deal with the inter-element discontinuities, leading to more complicated theoretical analysis. Secondly, in most previous stability analysis of LDG methods, a key step is to construct the relationship between the derivatives of the primitive variable and the auxiliary variables. This idea works for linear problems. However, our system is highly nonlinear and all the variables are coupled together. As an alternative, we will introduce a new auxiliary variable containing both the convection and diffusion terms. Thirdly, we have to control the change of the porosity during time evolution to obtain physically relevant numerical approximations and uniform upper bounds. Fourthly, to handle the time level mismatch of the spatial discretization due to the time integrations, we will construct a special second-order time method. Finally, to handle the complexity due to the Forchheimer term, we extrapolate some non-essential variables to linearize the coupled system, avoiding complicated iterations. To the best knowledge of the authors, this is the first scheme with time accuracy greater than one discussing stability for wormhole propagations. Moreover, we will prove the optimal error estimates of the schemes under mild time step restrictions. Numerical experiments are also given to verify the theoretical results.

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Journal of Computational and Applied Mathematics