Splitting Methods for Partial Differential Equations with Rough Solutions

Holden, Helge; Karlsen, Kenneth H.; Lie, Knut–Andreas; Risebro, Nils Henrik

doi:10.4171/078

Cited by 154 publications

(138 citation statements)

References 129 publications

(193 reference statements)

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“…The convergence of the "outer" iteration {u (k) } to u* of the Lagged Diffusivity procedure involves solution for an unknown vector u of the matrix equation (17). This linear system may be solved by an operator splitting method.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Lagged diffusivity method for the solution of nonlinear diffusion convection problems with finite differences

Galligani

2012

Adv Differ Equ

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This article concerns with the analysis of an iterative procedure for the solution of a nonlinear nonstationary diffusion convection equation in a two-dimensional bounded domain supplemented by Dirichlet boundary conditions. This procedure, denoted Lagged Diffusivity method, computes the solution by lagging the diffusion term. A model problem is considered and a finite difference discretization for that model is described. Furthermore, properties of the finite difference operator are proved. Then, a sufficient condition for the convergence of the Lagged Diffusivity method is given. At each stage of the iterative procedure, a linear system has to be solved and the Arithmetic Mean method is used. Numerical experiments show the efficiency, for different test functions, of the Lagged Diffusivity method combined with the Arithmetic Mean method as inner solver. Better results are obtained when the convection term increases. Statement of the problemConsider as model problem the nonlinear diffusion convection equationwhere u = u(x, y, t) is the density function at the point (x, y) at the time t of a diffusion medium R, s = s(x, y, u) > 0 is the diffusion coefficient or diffusivity and is dependent on the solution u, a = a(x, y) ≥ 0 is the absorption term,ṽ =ṽ(x, y, t, u) is the velocity vector and the source term s(x, y, t) is a real valued sufficiently smooth function.Equation (1) can be supplemented by the initial condition (t = 0)in the closureR of R and by Dirichlet boundary condition on the contour ∂R of R of the formIn the following, we suppose R to be a rectangular domain with boundary ∂R and we assume that the functions s, a, and s satisfy the "smoothness" conditions:

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Section: Numerical Experimentsmentioning

confidence: 99%

Lagged diffusivity method for the solution of nonlinear diffusion convection problems with finite differences

Galligani

2012

Adv Differ Equ

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“…Various simple 2D Riemann problems are simulated with both schemes and the time-evolution of resulting L 1 errors are displayed in §4.1. A "fingering test" taken from [25] is presented in §4.2. More involved 2D Riemann problems taken from both [53] and [28] are studied in §4.3; in particular, the exact solution given in [22] is used in order to scrutinize the onset of pointwise errors in both schemes, see Fig.…”

Section: Organization Of the Papermentioning

confidence: 99%

A Two-Dimensional Version of the Godunov Scheme for Scalar Balance Laws

Gosse¹

2014

SIAM J. Numer. Anal.

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Abstract. A Godunov scheme is derived for two-dimensional scalar conservation laws without or with source terms following ideas originally proposed by Boukadida and LeRoux [6] in the context of a staggered Lax-Friedrichs scheme. In both situations, the numerical fluxes are obtained at each interface of a uniform Cartesian computational grid just by means of the "external waves" involved in the entropy solution of the elementary 2d Riemann problems; in particular, all the wave-interaction phenomena is overlooked. This restriction of the wave pattern suffices for deriving the exact numerical fluxes of the staggered LxF scheme, but it furnishes only an approximation for the Godunov scheme: we show that under convenient assumptions, these flux functions are smooth and the resulting discretization process is stable under nearly-optimal CFL restriction. A well-balanced extension is presented, relying on the Curl-free component of the Helmholtz decomposition of the source term. Several numerical tests against exact 2D solutions are performed for convex, non-convex and inhomogeneous equations and the time evolution of the L 1 truncation error is displayed.

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“…Let H(t) be the operator that advances the system (57) forward in time, and let S(t) be the corresponding stiff ODE operator for the system (58). Then we may consider two main classes of splitting methods [18]:…”

Section: Hyperbolic Relaxation Systemsmentioning

confidence: 99%

An exponential time-differencing method for monotonic relaxation systems

Aursand

Evje

Flåtten

et al. 2014

Applied Numerical Mathematics

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Abstract. We present first and second-order accurate exponential time differencing methods for a special class of stiff ODEs, denoted as monotonic relaxation ODEs. Some desirable accuracy and robustness properties of our methods are established. In particular, we prove a strong form of stability denoted as monotonic asymptotic stability, guaranteeing that no overshoots of the equilibrium value are possible. This is motivated by the desire to avoid spurious unphysical values that could crash a large simulation.We present a simple numerical example, demonstrating the potential for increased accuracy and robustness compared to established Runge-Kutta and exponential methods. Through operator splitting, an application to granulargas flow is provided.

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Splitting Methods for Partial Differential Equations with Rough Solutions

Cited by 154 publications

References 129 publications

Lagged diffusivity method for the solution of nonlinear diffusion convection problems with finite differences

Lagged diffusivity method for the solution of nonlinear diffusion convection problems with finite differences

A Two-Dimensional Version of the Godunov Scheme for Scalar Balance Laws

An exponential time-differencing method for monotonic relaxation systems

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