Abstract
In this paper we propose a time-splitting method for degenerate convection–diffusion equations perturbed stochastically by white noise. This work generalizes previous results on splitting operator techniques for stochastic hyperbolic conservation laws for the degenerate parabolic case. The convergence in
Funding statement: This work was partially supported by Mexico CONACyT Project CB2016-286437.
A Numerical methods
A.1 Switch flux-limiter scheme
To solve numerically the deterministic parabolic equation (1.5) with an initial condition u(x, 0) = u0(x), for x ∈ [a, b] ∈ ℝ, we consider a uniform grid for the spatial-time domain. Denoting by (xj, tn) the mesh point with tn = n(Δt) and xj = a + jΔx, n, j ∈ ℕ, being a mesh cell
where
with
The associated weight function is
by measuring the smoothness of the advective flux function with
Finally, an important term of this scheme is the switch function
which turns on and off the upwind flux, modifying the algorithm in presence of discontinuous or smooth regions of the solution. In [10], authors proved that the viscous Switch flux-limiter scheme (A.1) for equation (1.5) is TVD-stable and convergent if
with |
A.2 Linear Steklov method
Such iterative method was proposed in [11] for solving the stochastic differential equation (1.6). Discretizing the time domain [0, T] with a constant time-step Δt such that tn = nΔt for some n ∈ ℕ. Denoting by Un the approximation and Wn the discrete standard Brownian motion satisfying
The Itô integral is approximated by
Assuming that the drift function g(u) can be linearized of the following form
where a1 is a scalar function, then the linear Steklov iterative process is written as
With regular conditions on local Lipschitz functions g and h satisfying a monotony property, the linear Steklov method becomes a robust and strongly convergent algorithm with a standard order of one-half.
References
[1] C. Bauzet, G. Vallet, and P. Wittbold, A degenerate parabolic–hyperbolic Cauchy problem with a stochastic force, J. Hyperbol. Differ. Equ., 12 (2015), 501–533.10.1142/S0219891615500150Search in Google Scholar
[2] C. Bauzet, Time-splitting approximation of the Cauchy problem for a stochastic conservation law, Math. Comput. Simulat., 118 (2015), 73–86.10.1016/j.matcom.2014.11.012Search in Google Scholar
[3] R. Bürger, R. Ruiz, K. Schneider, and M. Sepúlveda, Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension, ESAIM-Math. Model. Num., 42 (2008), 535–563.10.1051/m2an:2008016Search in Google Scholar
[4] R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Ration. Mech. Anal., 88 (1985), 223–270.10.1007/BF00752112Search in Google Scholar
[5] S. Evje and K. H. Karslen, Monotone Difference Approximations of BV Solutions to Degenerate Convection–Diffusion Equations, SIAM J. Numer. Anal., 37 (2000), 1838–1860.10.1137/S0036142998336138Search in Google Scholar
[6] S. Evje, K. H. Karlsen, K. A. Lie, and N. H. Risebro, Front tracking and operator splitting for nonlinear degenerate convection–diffusion equations. In: Parallel Solution of Partial Differential Equations, Springer, New York, 2000, pp. 209–227.10.1007/978-1-4612-1176-1_9Search in Google Scholar
[7] R. Eymard, T. Gallouët, and R. Herbin, Existence and uniqueness of the entropy solution to a nonlinear hyperbolic equation, Chinese Ann. Math. Ser. B, 16 (1995), 1–14.Search in Google Scholar
[8] H. Holden and N. H. Risebro, Conservation laws with a random source, Appl. Math. Optim., 36 (1997), 229–241.10.1007/s002459900061Search in Google Scholar
[9] M. Hutzenthaler, A. Jentzen, and P. E. Kloeden, Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. Royal Soc. London Ser. A, 467 (2011), 1563–1576.10.1098/rspa.2010.0348Search in Google Scholar
[10] S. Jerez and M. Arciga, Switch flux limiter method for viscous and nonviscous conservation laws, Appl. Math. Comput., 246 (2014), 292–305.10.1016/j.amc.2014.08.011Search in Google Scholar
[11] S. Jerez and S. Díaz-Infante, Exact difference schemes for stochastic differential equations. In: Exact Finite-Difference Schemes, De Gruyter, 2016, pp. 204–218.10.1515/9783110491326-008Search in Google Scholar
[12] S. Jerez and C. Parés, Entropy stable schemes for degenerate convection–diffusion equations, SIAM J. Numer. Anal., 55 (2017), 240–264.10.1137/16M1076411Search in Google Scholar
[13] K. H. Karlsen and N. H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients, Discrete Contin. Dyn. Syst. Ser. A, 9 (2003), 1081–1104.10.3934/dcds.2003.9.1081Search in Google Scholar
[14] J. D. R. Kersner, On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium, J. Differ. Equations, 69 (1987), 368–403.10.1016/0022-0396(87)90125-2Search in Google Scholar
[15] Y. Liu, C. W. Shu, and M. Zhang, High order finite difference WENO schemes for nonlinear degenerate parabolic equations, SIAM J. Sci. Comput., 33 (2011), 939–965.10.1137/100791002Search in Google Scholar
[16] J. D. Logan, Transport Modeling in Hydrogeochemical Systems, Springer, New York, 2001.10.1007/978-1-4757-3518-5Search in Google Scholar
[17] A. Michel and J. Vovelle, Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods, SIAM J. Numer. Anal., 41 (2003), 2262–2293.10.1137/S0036142902406612Search in Google Scholar
[18] F. Peyroutet, Etude d’une méthode Splitting pour des lois de conservation scalaires avec terme de source, Ph.D. thesis, U.P.P.A., 1999.Search in Google Scholar
[19] J. Proft and B. Riviere, Discontinuous Galerkin methods for convection–diffusion equations for varying and vanishing diffusivity, Int J. Numer. Anal. Mod., 6 (2009), 533–561.Search in Google Scholar
[20] G. Vallet, Uniqueness of the measure-valued solution to a conservation law with boundary conditions, Appl. Math. Lett., 11 (1998), 91–94.10.1016/S0893-9659(98)00062-7Search in Google Scholar
[21] G. Vallet, Stochastic perturbation of nonlinear degenerate parabolic problems, Differ. Integral Equ., 21 (2008), 1055–1082.Search in Google Scholar
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