Multirate Numerical Integration for Parabolic PDEs

Savcenco, Valeriu; Savcenco, E Eugeniu

doi:10.1063/1.2990964

Cited by 11 publications

(6 citation statements)

References 8 publications

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“…Combined space-time adaptivity, where ∆t is adapted also depending on the spatial location, are not easily accommodated by classical time stepping schemes, although some studies on local time stepping have appeared, see e.g. [EL94,FNWW09,Sav08].…”

mentioning

confidence: 99%

N-TERM WIENER CHAOS APPROXIMATION RATES FOR ELLIPTIC PDEs WITH LOGNORMAL GAUSSIAN RANDOM INPUTS

Hoáng

Schwab

2014

Math. Models Methods Appl. Sci.

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We consider diffusion in a random medium modeled as diffusion equation with lognormal Gaussian diffusion coefficient. Sufficient conditions on the log permeability are provided in order for a weak solution to exist in certain Bochner–Lebesgue spaces with respect to a Gaussian measure. The stochastic problem is reformulated as an equivalent deterministic parametric problem on ℝℕ. It is shown that the weak solution can be represented as Wiener–Itô Polynomial Chaos series of Hermite Polynomials of a countable number of i.i.d standard Gaussian random variables taking values in ℝ1. We establish sufficient conditions on the random inputs for weighted sequence norms of the Wiener–Itô decomposition coefficients of the random solution to be p-summable for some 0 < p < 1. For random inputs with additional spatial regularity, stronger norms of the weighted coefficient sequence in the random solutions' Wiener–Itô decomposition are shown to be p-summable for the same value of 0 < p < 1. We prove rates of nonlinear, best N-term Wiener Polynomial Chaos approximations of the random field, as well as of Finite Element discretizations of these approximations from a dense, nested family V0 ⊂ V1 ⊂ V2 ⊂ ⋯ V of finite element spaces of continuous, piecewise linear Finite Elements.

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mentioning

confidence: 99%

N-TERM WIENER CHAOS APPROXIMATION RATES FOR ELLIPTIC PDEs WITH LOGNORMAL GAUSSIAN RANDOM INPUTS

Hoáng

Schwab

2014

Math. Models Methods Appl. Sci.

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“…. Comparing equations (17) and (20), we can conclude that for any given m ≥ 1, the speed-up obtained from the compound-fast MRT method is always smaller than that obtained from the corresponding semi-implicit MRT method. However, for the limiting case of m → ∞, the speed-up from the compound-fast method approaches the speed-up from a stable semi-implicit method.…”

Section: Speed-upmentioning

confidence: 92%

“…The concept of MRT methods was introduced for systems of differential equations (ODEs and DAEs) in such studies as [9,10,16], and some recent results are presented in [2,4,7,14,18]. MRT methods for hyperbolic conservative laws are developed in [5,6,22] and for parabolic equations in [19,20]. A review of the MRT methods developed over the last two decades can be found in [8].…”

Section: Introductionmentioning

confidence: 99%

Multi-rate time stepping schemes for hydro-geomechanical model for subsurface methane hydrate reservoirs

Gupta

Wohlmuth

Helmig

2016

Advances in Water Resources

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We present an extrapolation-based semi-implicit multirate time stepping (MRT) scheme and a compound-fast MRT scheme for a naturally partitioned, multi-time-scale hydro-geomechanical hydrate reservoir model. We evaluate the performance of the two MRT methods compared to an iteratively coupled solution scheme and discuss their advantages and disadvantages. The performance of the two MRT methods is evaluated in terms of speed-up and accuracy by comparison to an iteratively coupled solution scheme. We observe that the extrapolation-based semi-implicit method gives a higher speed-up but is strongly dependent on the relative time scales of the latent (slow) and active (fast) components. On the other hand, the compound-fast method is more robust and less sensitive to the relative time scales, but gives lower speed up as compared to the semi-implicit method, especially when the relative time scales of the active and latent components are comparable.

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“…As a first example, we solve the following nonlinear reaction-diffusion equation, taken from [Savcenco et al 2005]:…”

Section: A Nonlinear Reaction-diffusion Equationmentioning

confidence: 99%

“…, N , of the approximate solution U ≈ u of the initial value problem (1). For related work on local time-stepping, see also [Hughes et al 1983a;1983b;Makino and Aarseth 1992;Davé et al 1997;Alexander and Agnor 1998;Osher and Sanders 1983;Flaherty et al 1997;Dawson and Kirby 2001;Lew et al 2003;Engstler and Lubich 1997;Savcenco et al 2005;Savcenco 2008]. In comparison with existing method for local time-stepping, the main advantage of the multi-adaptive Galerkin methods mcG(q) and mdG(q) is the automatic local step size selection based on a global a posteriori error estimate built into these methods.…”

Section: Introductionmentioning

confidence: 99%

Algorithms and Data Structures for Multi-Adaptive Time-Stepping

Jansson

Logg

2008

ACM Trans. Math. Softw.

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Multi-adaptive Galerkin methods are extensions of the standard continuous and discontinuous Galerkin methods for the numerical solution of initial value problems for ordinary or partial differential equations. In particular, the multi-adaptive methods allow individual and adaptive time steps to be used for different components or in different regions of space. We present algorithms for efficient multi-adaptive time-stepping, including the recursive construction of time slabs and adaptive time step selection. We also present data structures for efficient storage and interpolation of the multi-adaptive solution. The efficiency of the proposed algorithms and data structures is demonstrated for a series of benchmark problems.Comment: ACM Transactions on Mathematical Software 35(3), 24 pages (2008

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Multirate Numerical Integration for Parabolic PDEs

Cited by 11 publications

References 8 publications

N-TERM WIENER CHAOS APPROXIMATION RATES FOR ELLIPTIC PDEs WITH LOGNORMAL GAUSSIAN RANDOM INPUTS

N-TERM WIENER CHAOS APPROXIMATION RATES FOR ELLIPTIC PDEs WITH LOGNORMAL GAUSSIAN RANDOM INPUTS

Multi-rate time stepping schemes for hydro-geomechanical model for subsurface methane hydrate reservoirs

Algorithms and Data Structures for Multi-Adaptive Time-Stepping

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