## Mathematics and numerics for balance partial differential-algebraic equations (PDAEs).(English)Zbl 1440.76146

Summary: We study systems of partial differential-algebraic equations (PDAEs) of first order. Classical solutions of the theory of hyperbolic partial differential equation such as discontinuities (shock and contact discontinuities), rarefactions and diffusive traveling waves are extended for variables living on a surface $$\mathcal{S}$$, which is defined as solution of a set of algebraic equations. We propose here an alternative formulation to study numerically and theoretically the PDAEs by changing the algebraic conditions into partial differential equations with relaxation source terms (PDREs). The solution of such relaxed systems is proved to tend to the surface $$\mathcal{S}$$, i.e., to satisfy the algebraic equations for long times. We formulate a unified numerical scheme for systems of PDAEs and PDREs. This scheme is naturally parallelizable and has faster convergence. We do not perform a rigorous analysis about the convergence or accuracy for the method, the evidence of its effectiveness is presented by means of simulations for physical and synthetical problems.

### MSC:

 76S05 Flows in porous media; filtration; seepage 76M10 Finite element methods applied to problems in fluid mechanics 76M20 Finite difference methods applied to problems in fluid mechanics

### Software:

FISH; HYDROGEOCHEM; GEMSelektor; GEMS3K; HE-E1GODF; BCYCLIC; mctoolbox
Full Text:

### References:

 [1] Abgrall, R., Shu, C.W. (eds.): Handbook of Numerical Methods for Hyperbolic Problems. Applied and Modern Issues, vol. 18 (2017) · Zbl 1364.65001 [2] Abreu, E.; Colombeau, M.; Panov, EY, Weak asymptotic methods for scalar equations and systems, J. Math. Anal. Appl., 444, 1203-1232 (2016) · Zbl 1347.35081 [3] Abreu, E.; Bustos, A.; Ferraz, P.; Lambert, W., A relaxation projection analytical-numerical approach in hysteretic two-phase flows in porous media, J. Sci. Comput., 79, 9, 1936-1980 (2019) · Zbl 1418.76051 [4] Abreu, E.; Pérez, J., A fast, robust, and simple Lagrangian-Eulerian solver for balance laws and applications, Comput. Math. Appl., 77, 9, 2310-2336 (2019) [5] Abreu, E., Santo, A.E., Lambert, W., Perez, J.: Convergence of a Lagragian-Eulerian method via the weak asymptotic method. Submitted (2020) [6] Alvarez, A.C., Blom, T., Lambert, W.J., Bruining, J., Marchesin, D.: Low salinity carbonated waterflooding. In: ECMOR XV—15th European Conference on the Mathematics of Oil Recovery (2016) [7] Alvarez, AC; Blom, T.; Lambert, WJ; Bruining, J.; Marchesin, D., Analytical and numerical validation of a model for flooding by saline carbonated water, J. Petrol. Sci. Eng., 167, 900-917 (2018) [8] Alvarez, A.C., Bruining, J., Lambert, W.J., Marchesin, D.: The Riemann solution for carbonated waterflooding. In: ECMOR XV—15th European Conference on the Mathematics of Oil Recovery (2016) · Zbl 1405.65108 [9] Alvarez, AC; Bruining, J.; Lambert, WJ; Marchesin, D., Analytical and numerical solutions for carbonated waterflooding, Comput. Geosci., 22, 2, 505-526 (2018) · Zbl 1405.65108 [10] Alvarez, A.C., Goedert, G.T., Marchesin, D.: Resonance in rarefaction and shock curves: local analysis and numerics of the continuation method. arXiv preprint arXiv:1902.04182 (2019) [11] Antonietti, P.F., Cangiani, A., Collis, J., Dong, Z., Georgoulis, E.H., Giani, S., Houston, P.: Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains, pp. 281-310. Springer, Berlin (2016) · Zbl 1357.65251 [12] Arbenz, P.; Cleary, A.; Dongarra, J.; Hegland, M.; Amestoy, P.; Berger, P.; Daydé, M.; Ruiz, D.; Duff, I.; Frayssé, V.; Giraud, L., A comparison of parallel solvers for diagonally dominant and general narrow-banded linear systems II, Euro-Par’99 Parallel Processing, 1078-1087 (1999), Berlin: Springer, Berlin [13] Arnold, DN; Brezzi, F.; Cockburn, B.; Marini, LD, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 5, 1749-1779 (2002) · Zbl 1008.65080 [14] Atkinson, KE, An Introduction to Numerical Analysis (1978), New York: Wiley, New York [15] Azevedo, AV; Marchesin, D., Multiple viscous solutions for systems of conservation laws, Trans. Am. Math. Soc., 347, 8, 3061-3077 (1995) · Zbl 0851.35084 [16] Bartel, A.; Gunther, M., Pdaes in refined electrical network modeling, SIAM Rev., 60, 1, 56-91 (2018) · Zbl 1382.65282 [17] Barth, T., Herbin, R., Ohlberger, M.: Finite Volume Methods: Foundation and Analysis, pp. 1-60. American Cancer Society, New York (2017) [18] Benhammouda, B.; Vazquez-Leal, H., Analytical solutions for systems of partial differential algebraic equations, Springer Plus, 3, 137, 9 (2014) [19] Bondeli, S., Divide and conquer: a parallel algorithm for the solution of a tridiagonal linear system of equations, Parallel Comput., 17, 4, 419-434 (1991) · Zbl 0739.65016 [20] Bondeli, S.; Gander, W., Cyclic reduction for special tridiagonal systems, SIAM J. Matrix Anal. Appl., 15, 01 (1994) · Zbl 0806.65024 [21] Brenner, S.; Scott, RL, The Mathematical Theory of Finite Element Methods (2005), Berlin: Springer, Berlin [22] Chavez, G.; Turkiyyah, G.; Zampini, S.; Ltaief, H.; Keyes, D., Accelerated cyclic reduction: a distributed-memory fast solver for structured linear systems, Parallel Comput., 74, 65-83 (2018) [23] Chudej, K.; Pesch, HJ; Rang, J., Chapter: Index Analysis of Models of the Book Molten Carbonate Fuel Cells (2007), Weinheim: Wiley, Weinheim [24] Ciarlet, PG, The Finite Element Method for Elliptic Problems (2002), Philadelphia: SIAM, Philadelphia [25] Cockburn, B., An introduction to the discontinuous Galerkin method for convection-dominated problems, Adv. Numer. Approx. Nonlinear Hyperbolic Equ., 1697, 151-268 (1998) · Zbl 0927.65120 [26] Cockburn, B., Discontinuous Galerkin methods for convection dominated problems, High Order Methods Comput. Phys., 10, 69-224 (1999) · Zbl 0937.76049 [27] Dafermos, C., Hyperbolic Conservation Laws in Continuum Physics (2010), New York: Springer, New York · Zbl 1196.35001 [28] Delestre, O., Ghigo, A., Fullana, J.-M., Lagrée, P.-Y.: A shallow water with variable pressure model for blood flow simulation. arXiv preprint arXiv:1509.01917 (2015) · Zbl 1352.35200 [29] Delestre, O.; Lagrée, P-Y, A well-balanced finite volume scheme for blood flow simulation, Int. J. Numer. Methods Fluids, 72, 2, 177-205 (2013) [30] Demmel, JW; Higham, JH; Schreiber, RS, Stability of block LU factorization, Numer. Linear Algebra Appl., 2, 173-190 (1995) [31] Engesgaard, P.; Kipp, KL, A geochemical transport model for redox-controlled movement of mineral fronts in groundwater flow systems: a case of nitrate removal by oxidation of pyrite, Water Resour. Res., 28, 2829-2843 (1992) [32] Evans, LC, Partial Differential Equations (2010), Providence: American Mathematical Society, Providence [33] Eymard, R., Gallouët, T., Herbin, R.: Finite Volume Methods. J. L. Lions; Philippe Ciarlet. Solution of Equation in R3 (Part 3). Tech. Sci. Comput. (Part 3) 7, 713-1020 (2000) [34] Fjordholm, US; Mishra, S.; Tadmor, E., Arbitrary order accurate essentially non-oscillatory entropy stable schemes for systems of conservation laws, SIAM J. Numer. Anal, 2, 50, 544-573 (2012) · Zbl 1252.65150 [35] Fuchs, F.; McMurry, A.; Mishra, S.; Risebro, NH; Waagan, K., Approximate Riemann solver based high-order finite volume schemes for the MHD equations in multidimensions, Commun. Comput. Phys, 9, 324-362 (2011) · Zbl 1364.76113 [36] Fuhrer, C.; Rannacher, R., An adaptive streamline diffusion finite element method for hyperbolic conservation laws, J. Numer. Math., 5, 1, 145-162 (1997) · Zbl 0885.65107 [37] Girault, V.; Raviart, PA, Finite Element Approximation of the Navier-Stokes Equations. Lecture Notes in Mathematics (1979), Berlin: Springer, Berlin · Zbl 0413.65081 [38] Glowinski, R., Numerical Methods for Nonlinear Variational Problems (1984), Berlin: Springer, Berlin · Zbl 0575.65123 [39] Godlewski, E.; Raviart, PA, Numerical Approximation of Hyperbolic Systems of Conservation Laws (1996), Berlin: Springer, Berlin [40] Gottlieb, D., Hesthaven, J.S.: Spectral Methods for Hyperbolic Problems. Numerical Analysis 2000, vol. 7. Elsevier, Amsterdam (2001) · Zbl 0974.65093 [41] Gottlieb, S.; Shu, C-W; Tadmor, E., High order time discretizations with strong stability properties, SIAM. Rev., 43, 89-112 (2001) [42] Gunther, M.; Wagner, Y., Index concepts for linear mixed systems of differential-algebraic and hyperbolic-type equations, SIAM J. Sci. Comput., 22, 5, 1610-1629 (2000) · Zbl 0981.65110 [43] Hansbo, P., The characteristic streamline diffusion method for the time-dependent incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 99, 1, 171-186 (1992) · Zbl 0825.76423 [44] Harten, A.; Engquist, B.; Osher, S.; Chakravarty, SR, Uniformly high order accurate essentially non-oscillatory schemes. III, J. Comput. Phys., 2, 71, 231-303 (1987) · Zbl 0652.65067 [45] Hesthaven, JS; Gottlieb, S.; Gottlieb, D., Spectral Methods for Time-Dependent Problems (2007), Cambridge: Cambridge University Press, Cambridge · Zbl 1111.65093 [46] Higham, NJ, Accuracy and Stability of Numerical Algorithms. Other Titles in Applied Mathematics (2002), Philadelphia: Society for Industrial and Applied Mathematics, Philadelphia [47] Hime, G.: Parallel Solution of Nonlinear Balance Systems. Master’s thesis, Laboratorio Nacional de Computacao Cientifica, Petropolis (2007) [48] Hirshman, SP; Perumalla, S.; Lynch, VE; Sanchez, R., Bcyclic: a parallel block tridiagonal matrix cyclic solver, J. Comput. Phys., 229, 6392-6404 (2010) · Zbl 1197.65032 [49] Houston, P.; Schwab, C.; Süli, E., Stabilized $$hp$$-finite element methods for first-order hyperbolic problems, SIAM J. Numer. Anal., 37, 5, 1618-1643 (2000) · Zbl 0957.65103 [50] Houston, P.; Schwab, C.; Süli, E., Discontinuous $$hp$$-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 39, 6, 2133-2163 (2002) · Zbl 1015.65067 [51] Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite-Element Analysis. Dover Civil and Mechanical Engineering, 1st edn (2000) · Zbl 1191.74002 [52] Hughes, TJR; Brooks, A., Streamline upwind/Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 32, 1, 199-259 (1982) · Zbl 0497.76041 [53] Izadi, M., Streamline diffusion method for treating coupling equations of hyperbolic scalar conservation laws, Math. Comput. Model., 45, 1, 201-214 (2007) · Zbl 1133.65076 [54] Jiang, G-S; Shu, C-W, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 1, 126, 202-228 (1996) · Zbl 0877.65065 [55] Jin, S.; Xin, P., The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Commun. Pure Appl. Math., 48, 3, 253-281 (1995) [56] John, F., Partial Differential Equations (1971), Berlin: Springer, Berlin [57] Johnson, C., Numerical Solution of Partial Differential Equations by the Finite Element Method (1988), New York: Cambridge University Press, New York [58] Kappeli, R.; Whitehouse, SC; Scheidegger, S.; Pen, U-L; Liebendörfer, M., Fish: a three-dimensional parallel magnetohydrodynamics code for astrophysical applications, Astrophys. J. Suppl., 195, 20 (2011) [59] Kroner, D., Numerical Schemes for Conservation Laws (1997), Leipzig: Wiley Teubner, Leipzig · Zbl 0872.76001 [60] Kulik, DA; Wagner, T.; Dmytrieva, SV; Kosakowski, G.; Hingerl, FF; Chudnenko, KV; Berner, UR, GEM-Selektor geochemical modeling package: revised algorithm and GEMS3K numerical kernel for coupled simulation codes, Comput. Geosci., 17, 1, 1-24 (2013) · Zbl 1356.86022 [61] Lambert, W.J., Alvarez, A.C., Marchesin, D., Bruining, J.: Mathematical theory of two-phase geochemical flow with chemical species. In: Klingenberg, C., Westdickenberg, M. (eds.) Theory, Numerics and Applications of Hyperbolic Problems II. HYP 2016. Springer Proceedings in Mathematics and Statistics, vol. 237, pp. 255-267 (2018) · Zbl 1407.76173 [62] Lambert, WJ; Alvarez, AC; Matos, V.; Marchesin, D.; Bruining, J., Nonlinear wave analysis of geochemical injection for multicomponent two phase flow in porous media, J. Differ. Equ., 266, 406-454 (2019) · Zbl 1406.35284 [63] Lax, PD; Richtmyer, RD, Survey of the stability of linear finite difference equations, Commun. Pure Appl. Math., 9, 2, 267-293 (1956) · Zbl 0072.08903 [64] Leveque, R.J.: Numerical Methods for Conservation Laws. Lecture Notes in Mathematics Springer Basel AG (1990) · Zbl 0723.65067 [65] Leveque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics, Cambridge (2002) · Zbl 1010.65040 [66] Liu, TP, The Riemann problem for general $$2 \times 2$$ conservation laws, Trans. Am. Math. Soc., 199, 89-112 (1974) · Zbl 0289.35063 [67] Liu, T-P, The Riemann problem for general systems of conservation laws, J. Differ. Equ., 18, 1, 218-234 (1975) · Zbl 0297.76057 [68] Liu, Y-J; Shu, C-W; Tadmor, E.; Zhang, M., Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction, SIAM J. Numer. Anal., 45, 6, 2442-2467 (2007) · Zbl 1157.65450 [69] Marszalek, W.: Analysis of Partial Differential Algebraic Equations. Master’s thesis, North Carolina State University, Raleigh (1997) [70] Martinson, WS; Barton, PI, A differentiation index for partial differential-algebraic equations, SIAM J. Sci. Comput., 21, 6, 2295-2315 (2000) · Zbl 0956.35026 [71] Mattheij, RMM, The stability of LU-decompositions of block tridiagonal matrices, Bull. Aust. Math. Soc., 29, 4 (1984) [72] Mehrmann, V., Divide and conquer methods for block tridiagonal systems, Parallel Comput., 19, 257-279 (1993) · Zbl 0765.65035 [73] Muniruzzamana, M.; Rollea, M., Modeling multicomponent ionic transport in groundwater with IPhreeqc coupling: electrostatic interactions and geochemical reactions in homogeneous and heterogeneous domains, Adv. Water Resour., 98, 1-15 (2016) [74] Olufsen, MS; Peskin, CS; Kim, WY; Pedersen, EM; Nadim, A.; Larsen, J., Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions, Ann. Biomed. Eng., 28, 11, 1281-1299 (2000) [75] Raja Sekhar, T., Minhajul: Elementary wave interactions in blood flow through artery. J. Math. Phys. 58(10), 101502 (2017) · Zbl 1374.76242 [76] Raviart, P.-A., Thomas, J.-M.: Introduction l’analyse numrique des equations aux drives partielles. Collection Mathématiques Appliques pour la Matrise, Masson (1983) [77] Reintjes, M., Constrained systems of conservation laws: a geometric theory, Methods Appl. Anal., 24, 4, 407-404 (2017) [78] Serre, D., Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves (1999), Cambridge: Cambridge University Press, Cambridge [79] Serre, D., Systems of Conservation Laws, 2 volumes (1999), England: Cambridge University Press, England [80] Smoller, J., Shock Waves and Reaction-Diffusion Equations (1983), New York: Springer, New York · Zbl 0508.35002 [81] Sod, G.A.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27 (1978) · Zbl 0387.76063 [82] Sonnendrucker, E.: Finite Element Methods for Hyperbolic Systems. Lecture notes Wintersemester 2014-2015, Max Planck Institut (2015) [83] Strang, G.; Fix, GJ, An Analysis of the Finite Element Method (1973), Englewood Cliffs: Prentice-Hall Inc, Englewood Cliffs · Zbl 0278.65116 [84] Strikwerda, JC, Finite Difference Schemes and Partial Differential Equations (1989), Belmont: Wadsworth Publ. Co., Belmont [85] Szabo, B.; Babuska, I., Finite Element Method Analysis (1988), New York: Wiley, New York [86] Tadmor, E.: A review of numerical methods for nonlinear partial differential equations. Bull. (New Series) Am. Math. Soc. 49(4), 507-554 (2012) · Zbl 1258.65073 [87] Toro, EF, Riemann Solvers and Numerical Methods for Fluid Dynamics (A Practical Introduction) (2009), Berlin: Springer, Berlin [88] Varah, JM, On the solution of block-tridiagonal systems arising from certain finite-difference equations, Math. Comput., 26, 120, 859-868 (1972) · Zbl 0266.65029 [89] Wahanik, H.: Thermal Effects in the Injection of CO2 in Deep Underground Aquifers. Ph.D. thesis, IMPA, Rio de Janeiro (2011) [90] Wang, XY; Zhu, ZS; Lu, YK, Solitary wave solutions of the generalised Burgers-Huxley equation, J. Phys. A Math. Gen., 23, 3, 271-274 (1990) · Zbl 0708.35079 [91] Yeh, GT; Tripathi, VS, A critical evaluation of recent developments in hydrogeochemical transport models of reactive multichemical components, Water Resour. Res., 25, 1, 93-108 (1989) [92] Yeh, GT; Tripathi, VS, A model for simulating transport of reactive multispecies components: model development and demonstration, Water Resour. Res., 27, 12, 3075-3094 (1991) [93] Yeh, GT; Fang, Y.; Zhang, F.; Sun, J.; Li, Y.; Li, MH; Siegel, MD, Numerical modeling of coupled fluid flow and thermal and reactive biogeochemical transport in porous and fractured medias, Comput. Geosci., 14, 149-170 (2010) · Zbl 1398.76174
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.