Abstract
In this work, various high-accuracy numerical schemes for transport problems in fractured media are further developed and compared. Specifically, to capture sharp gradients and abrupt changes in time, schemes with low order of accuracy are not always sufficient. To this end, discontinuous Galerkin up to order two, Streamline Upwind Petrov-Galerkin, and finite differences, are formulated. The resulting schemes are solved with sparse direct numerical solvers. Moreover, time discontinuous Galerkin methods of order one and two are solved monolithically and in a decoupled fashion, respectively, employing finite elements in space on locally refined meshes. Our algorithmic developments are substantiated with one regular fracture network and several further configurations in fractured media with large parameter contrasts on small length scales. Therein, the evaluation of the numerical schemes and implementations focuses on three key aspects, namely accuracy, monotonicity, and computational costs.
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The data generated or analyzed during this work are available from the corresponding author on reasonable request.
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The authors gratefully acknowledge the financial support from the doctoral program “International Research Training Group (IRTG)” 2657 funded by the German Research Foundation (DFG).
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Insa Neuweiler and Thomas Wick are both contributed equally to this work.
Appendices
Appendix A Sequential solution of TDG
Saying we decompose the temporal domain T into \(\mathcal {T}_h\), which contains 3 subdomains: \(I_1, I_2\) and \(I_3\). On each subdomain we apply TDG(1), thus we have in total 6 temporal nodes. The block linear system Eq. 27 without jump terms then reads:
After adding jump terms, Eq. A1 becomes:
Apparently, \(\varvec{\xi }^1\) and \(\varvec{\xi }^2\) can directly be obtained by solving
After having \(\varvec{\xi }^2\), \(\varvec{\xi }^3\) and \(\varvec{\xi }^4\) can be obtained by solving
Similarly, we can solve \(\varvec{\xi }^5\) and \(\varvec{\xi }^6\) with \(\varvec{\xi }^4\).
Therefore, for an arbitrarily large number of time subintervals, we can always solve the block linear system sequentially in time. This property is quite favorable in practice for avoiding solving a huge block linear system, namely decreasing computation times.
Appendix B Solve block linear system
Using the sequential solution strategy (Appendix A), we finally need to solve a block linear system (e.g. Eqs. A3 or A4). The matrix on the left-hand side is a block sparse matrix (see Fig. 13). It is hence hard to find a good preconditioner matrix if one wants to use an iterative solver.
A fixed-point iteration method (see e.g. [65]) can be used to solve the block linear system in a decoupled way. The main two advantages are
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1.
the dimension of the original block linear system is decreased;
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2.
for every sublinear system the matrix on the left-hand side is a standard FE sparse matrix which is diagonally dense and non-diagonally sparse (see Fig. 14).
Thus the computation times can be greatly decreased.
We use the block linear system (A3) as the instance. First reformulate (A3) with the two sublinear systems:
Algorithm 3 describes the process of how to solve the two sublinear systems iteratively. We use the direct solver (UMFPACK) to solve the systems (A5) and (A6).
Similarly, we choose \(\varvec{\xi }^2\) as the initial guess of \(\varvec{\xi }^3\), when Eq. A4 is considered. Our experience is that the chosen initial guess obtained from the last time step is very close to the unknown vector considered in the current time step at the jump point. Because the amplitude of the jump term should not be very large. Hence, we can converge to the desired result in a few steps with a proper relative tolerance.
We remark that this method is not quite feasible for the case where the degree of polynomials in time is greater than 2. Nevertheless, since TDG(0) and TDG(1) are most widely used in practice, this method still has considerable value in application.
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Gao, W., Neuweiler, I. & Wick, T. A comparison study of spatial and temporal schemes for flow and transport problems in fractured media with large parameter contrasts on small length scales. Comput Geosci (2024). https://doi.org/10.1007/s10596-024-10293-y
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DOI: https://doi.org/10.1007/s10596-024-10293-y