Positivity preserving high-order local discontinuous Galerkin method for parabolic equations with blow-up solutions. (English) Zbl 1352.65345
Summary: In this paper, we apply positivity-preserving local discontinuous Galerkin (LDG) methods to solve parabolic equations with blow-up solutions. This model is commonly used in combustion problems. However, previous numerical methods are mainly based on a second order finite difference method. This is because the positivity-preserving property can hardly be satisfied for high-order ones, leading to incorrect blow-up time and blow-up sets. Recently, we have applied discontinuous Galerkin methods to linear hyperbolic equations involving {\(\delta\)}-singularities and obtained good approximations. For nonlinear problems, some special limiters are constructed to capture the singularities precisely. We will continue this approach and study parabolic equations with blow-up solutions. We will construct special limiters to keep the positivity of the numerical approximations. Due to the Dirichlet boundary conditions, we have to modify the numerical fluxes and the limiters used in the schemes. Numerical experiments demonstrate that our schemes can capture the blow-up sets, and high-order approximations yield better numerical blow-up time.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 35B44 | Blow-up in context of PDEs |
| 35K59 | Quasilinear parabolic equations |
Keywords:
positivity-preserving; blow-up solutions; local discontinuous Galerkin method; Dirichlet boundary conditionsReferences:
| [1] | Abia, L. M.; López-Marcos, J. C.; Martinez, J., On the blow-up time convergence of semidiscretizations of reaction-diffusion equations, Appl. Numer. Math., 26, 399-414 (1998) · Zbl 0929.65070 |
| [2] | Acosta, G.; Durán, R. G.; Rossi, J. D., An adaptive time step procedure for a parabolic problem with blow-up, Computing, 68, 343-373 (2002) · Zbl 1005.65102 |
| [3] | Bassi, F.; Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131, 267-279 (1997) · Zbl 0871.76040 |
| [4] | Bebernes, J.; Eberly, D., Mathematical Problems from Combustion Theory (1989), Springers-Verlag · Zbl 0692.35001 |
| [5] | Boni, T. K., On blow-up and asymptotic behavior of solutions to a nonlinear parabolic equation of second order, Asymptot. Anal., 21, 187-208 (1999) · Zbl 0945.35048 |
| [6] | Brezis, H., Blow-up of \(u_t = u_{x x} + g(u)\) revisited, Adv. Differ. Equ., 1, 73-90 (1996) · Zbl 0855.35063 |
| [7] | Bandle, C.; Brunner, H., Numerical analysis of semilinear parabolic problems with blow-up solutions, Rev. R. Acad. Cienc. Exactas Fís. Nat., 88, 203-222 (1994) · Zbl 0862.65053 |
| [8] | Bandle, C.; Brunner, H., Blowup in diffusion equations: a survey, J. Comput. Appl. Math., 97, 3-22 (1998) · Zbl 0932.65098 |
| [9] | Brändle, C.; Groisman, P.; Rossi, J. D., Fully discrete adaptive methods for a blow-up problem, Math. Models Methods Appl. Sci., 14, 1425-1450 (2004) · Zbl 1065.35022 |
| [10] | Brändle, C.; Quirós, F.; Rossi, J. D., An adaptive numerical method to handle blow-up in a parabolic system, Numer. Math., 102, 39-59 (2005) · Zbl 1082.65094 |
| [11] | Budd, C. J.; Huang, W.; Russell, R. D., Moving mesh methods for problems with blow-up, SIAM J. Sci. Comput., 17, 305-327 (1996) · Zbl 0860.35050 |
| [12] | Cheng, Y.; Shu, C.-W., A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives, Math. Comput., 77, 699-730 (2008) · Zbl 1141.65075 |
| [13] | Cho, C.-H., On the finite difference approximation for blow-up solutions of the porous medium equation with a source, Appl. Numer. Math., 65, 1-26 (2013) · Zbl 1312.76040 |
| [14] | Cockburn, B.; Hou, S.; Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comput., 54, 545-581 (1990) · Zbl 0695.65066 |
| [15] | Cockburn, B.; Lin, S.-Y.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, J. Comput. Phys., 84, 90-113 (1989) · Zbl 0677.65093 |
| [16] | Cockburn, B.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comput., 52, 411-435 (1989) · Zbl 0662.65083 |
| [17] | Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141, 199-224 (1998) · Zbl 0920.65059 |
| [18] | Cockburn, B.; Shu, C.-W., The local discontinuous Galerkin method for time dependent convection-diffusion systems, SIAM J. Numer. Anal., 35, 2440-2463 (1998) · Zbl 0927.65118 |
| [19] | Cockburn, B.; Dong, B., An analysis of the minimal dissipation local discontinuous Galerkin method for convection-diffusion problems, J. Sci. Comput., 32, 233-262 (2007) · Zbl 1143.76031 |
| [20] | Ferreira, R.; Groisman, P.; Rossi, J. D., Numerical blow-up for a nonlinear problem with a nonlinear boundary condition, Math. Models Methods Appl. Sci., 12, 461-483 (2002) · Zbl 1028.35082 |
| [21] | Galaktionov, V. A.; Vazquez, J. L., The problem of blow-up in nonlinear parabolic equation, Discrete Contin. Dyn. Syst., 8, 399-433 (2002) · Zbl 1010.35057 |
| [22] | Groisman, P., Totally discrete explicit and semi-implicit Euler methods for a blow-up problem in several space dimensions, Computing, 76, 325-352 (2006) · Zbl 1087.65093 |
| [23] | Gottlieb, S.; Shu, C.-W.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev., 43, 89-112 (2001) · Zbl 0967.65098 |
| [24] | Johnson, C.; Pitkäranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comput., 46, 1-26 (1986) · Zbl 0618.65105 |
| [25] | Li, F. S., Global existence and uniqueness of weak solution to nonlinear viscoelastic full Marguerre-von Kármán shallow equations, Acta Math. Sin. Engl. Ser., 25, 2133-2156 (2009) · Zbl 1423.74337 |
| [26] | Mai, I.; Mochizuki, K., On blow-up of solutions for quasilinear degenerate parabolic equations, Publ. Res. Inst. Math. Sci., 27, 695-709 (1991) · Zbl 0789.35022 |
| [27] | Nabongo, D.; Boni, T. K., Blow-up for semidiscretization of a localized semilinear heat equation, J. Appl. Anal., 2, 173-204 (2009) · Zbl 1190.35013 |
| [28] | N’Gohisse, F. K.; Boni, T. K., Numerical blow-up for a nonlinear heat equation, Acta Math. Sin. Engl. Ser., 27, 845-862 (2011) · Zbl 1221.35075 |
| [29] | Quittner, P.; Souplet, P., Superlinear Parabolic ProblemsBlow-up, Global Existence and Steady States, Birkhäuser Adv. Texts (2007), Birkhäuser: Birkhäuser Basel · Zbl 1128.35003 |
| [30] | Reed, W. H.; Hill, T. R., Triangular mesh methods for the Neutron transport equation (1973), Los Alamos Scientific Laboratory: Los Alamos Scientific Laboratory Los Alamos, NM, Report LA-UR-73-479 |
| [31] | Le Roux, M.-N., Semidiscretization in time of nonlinear parabolic equations with blow-up of the solution, SIAM J. Numer. Anal., 31, 170-195 (1994) · Zbl 0803.65095 |
| [32] | Shu, C.-W., Total-variation-diminishing time discretizations, SIAM J. Sci. Stat. Comput., 9, 1073-1084 (1988) · Zbl 0662.65081 |
| [33] | Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 439-471 (1988) · Zbl 0653.65072 |
| [34] | Suzuki, R., On blow-up sets and asymptotic behavior of interface of one dimensional quasilinear degenerate parabolic equation, Publ. Res. Inst. Math. Sci., 27, 375-398 (1991) · Zbl 0789.35024 |
| [35] | Ushijima, T. K., On the approximation of blow-up time for solutions of nonlinear parabolic equations, Publ. Res. Inst. Math. Sci., 36, 613-640 (2000) · Zbl 0981.65106 |
| [37] | Xu, Z., Parametrized maximum principle preserving flux limiters for high order schemes solving hyperbolic conservation laws: one-dimensional scalar problem, Math. Comput., 83, 2213-2238 (2014) · Zbl 1300.65063 |
| [38] | Yang, Y.; Shu, C.-W., Discontinuous Galerkin methods for hyperbolic equations involving \(δ\)-singularities: negative-order norm error estimates and applications, Numer. Math., 124, 753-781 (2013) · Zbl 1273.65152 |
| [39] | Yang, Y.; Wei, D.; Shu, C.-W., Discontinuous Galerkin method for Krause’s consensus models and pressureless Euler equations, J. Comput. Phys., 252, 109-127 (2013) · Zbl 1349.65497 |
| [40] | Zhao, X.; Yang, Y.; Seyler, C., A positivity-preserving semi-implicit discontinuous Galerkin scheme for solving extended magnetohydrodynamics equations, J. Comput. Phys., 278, 400-415 (2014) · Zbl 1349.76296 |
| [41] | Zhang, L.; Ziang, Y. S.; Zhou, Z., Parabolic equation with VMO coefficients in generalized Morrey, Acta Math. Sin. Engl. Ser., 26, 117-130 (2010) · Zbl 1191.35147 |
| [42] | Zhang, Q.; Wu, Z.-L., Numerical simulation for porous medium equation by local discontinuous Galerkin finite element method, J. Sci. Comput., 38, 127-148 (2009) · Zbl 1203.65193 |
| [43] | Zhang, X.; Shu, C.-W., On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comput. Phys., 229, 3091-3120 (2010) · Zbl 1187.65096 |
| [44] | Zhang, X.; Shu, C.-W., On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, J. Comput. Phys., 229, 8918-8934 (2010) · Zbl 1282.76128 |
| [45] | Zhang, Y.; Zhang, X.; Shu, C.-W., Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection-diffusion equations on triangular meshes, J. Comput. Phys., 234, 295-316 (2013) · Zbl 1284.65140 |
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