It is known that the solutions to space‐fractional diffusion equations exhibit singularities near the boundary. Therefore, numerical methods discretized on the composite mesh, in which the mesh size is refined near the boundary, provide more precise approximations to the solutions. However, the coefficient matrices of the corresponding linear systems usually lose the diagonal dominance and are ill‐conditioned, which in turn affect the convergence behavior of the iteration methods.In this work we study a finite volume method for two‐sided fractional diffusion equations, in which a locally refined composite mesh is applied to capture the boundary singularities of the solutions. The diagonal blocks of the resulting three‐by‐three block linear system are proved to be positive‐definite, based on which we propose an efficient block Gauss–Seidel method by decomposing the whole system into three subsystems with those diagonal blocks as the coefficient matrices. To further accelerate the convergence speed of the iteration, we use T. Chan's circulant preconditioner31as the corresponding preconditioners and analyze the preconditioned matrices' spectra. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method and its strong potential in dealing with ill‐conditioned problems. While we have not proved the convergence of the method in theory, the numerical experiments show that the proposed method is convergent.
Statistical solutions of hyperbolic systems of conservation laws: Numerical approximation
Statistical solutions are time-parameterized probability measures on spaces of integrable functions, which have been proposed recently as a framework for global solutions and uncertainty quantification for multi-dimensional hyperbolic system of conservation laws. By combining high-resolution finite volume methods with a Monte Carlo sampling procedure, we present a numerical algorithm to approximate statistical solutions. Under verifiable assumptions on the finite volume method, we prove that the approximations, generated by the proposed algorithm, converge in an appropriate topology to a statistical solution. Numerical experiments illustrating the convergence theory and revealing interesting properties of statistical solutions are also presented.
more »
« less
- Award ID(s):
- 1912854
- NSF-PAR ID:
- 10170130
- Date Published:
- Journal Name:
- Mathematical Models and Methods in Applied Sciences
- Volume:
- 30
- Issue:
- 03
- ISSN:
- 0218-2025
- Page Range / eLocation ID:
- 539 to 609
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
more » « less
Abstract -
A numerical and analytical study is made of the macroscopic or homogenized mechanical response of a random isotropic suspension of liquid n -spherical inclusions ( n = 2, 3), each having identical initial radius A , in an elastomer subjected to small quasistatic deformations. Attention is restricted to the basic case when the elastomer is an isotropic incompressible linear elastic solid, the liquid making up the inclusions is an incompressible linear elastic fluid, and the interfaces separating the solid elastomer from the liquid inclusions feature a constant initial surface tension γ . For such a class of suspensions, it has been recently established that the homogenized mechanical response is that of a standard linear elastic solid and hence, for the specific type of isotropic incompressible suspension of interest here, one that can be characterized solely by an effective shear modulus n in terms of the shear modulus μ of the elastomer, the initial elasto-capillary number eCa = γ /2 μA , the volume fraction c of inclusions, and the space dimension n . This paper presents numerical solutions—generated by means of a recently introduced finite-element scheme—for n over a wide range of elasto-capillary numbers eCa and volume fractions of inclusions c . Complementary to these, a formula is also introduced for n that is in quantitative agreement with all the numerical solutions, as well as with the asymptotic results for n in the limit of dilute volume fraction of inclusions and at percolation . The proposed formula has the added theoretical merit of being an iterated-homogenization solution.more » « less
-
Performing numerical integration when the integrand itself cannot be evaluated point-wise is a challenging task that arises in statistical analysis, notably in Bayesian inference for models with intractable likelihood functions. Markov chain Monte Carlo (MCMC) algorithms have been proposed for this setting, such as the pseudo-marginal method for latent variable models and the exchange algorithm for a class of undirected graphical models. As with any MCMC algorithm, the resulting estimators are justified asymptotically in the limit of the number of iterations, but exhibit a bias for any fixed number of iterations due to the Markov chains starting outside of stationarity. This "burn-in" bias is known to complicate the use of parallel processors for MCMC computations. We show how to use coupling techniques to generate unbiased estimators in finite time, building on recent advances for generic MCMC algorithms. We establish the theoretical validity of some of these procedures by extending existing results to cover the case of polynomially ergodic Markov chains. The efficiency of the proposed estimators is compared with that of standard MCMC estimators, with theoretical arguments and numerical experiments including state space models and Ising models.more » « less
-
null (Ed.)ABSTRACT We describe and test a family of new numerical methods to solve the Schrödinger equation in self-gravitating systems, e.g. Bose–Einstein condensates or ‘fuzzy’/ultra-light scalar field dark matter. The methods are finite-volume Godunov schemes with stable, higher order accurate gradient estimation, based on a generalization of recent mesh-free finite-mass Godunov methods. They couple easily to particle-based N-body gravity solvers (with or without other fluids, e.g. baryons), are numerically stable, and computationally efficient. Different sub-methods allow for manifest conservation of mass, momentum, and energy. We consider a variety of test problems and demonstrate that these can accurately recover solutions and remain stable even in noisy, poorly resolved systems, with dramatically reduced noise compared to some other proposed implementations (though certain types of discontinuities remain challenging). This is non-trivial because the ‘quantum pressure’ is neither isotropic nor positive definite and depends on higher order gradients of the density field. We implement and test the method in the code gizmo.more » « less
-
A novel numerical method is proposed for the solution of transient multi-physics problems involving heat conduction, electrical current sharing and Joule heating. The innovation consists of a mesh-free Monte Carlo approach that eliminates or drastically reduces the particle scattering requirements typical of conventional Monte-Carlo methods. The proposed algorithm encapsulates a volume around each point that affects the solution at a given point in the domain; the volume includes other points that represent small perturbations along the path of energy transfer. The proposed method is highly parallelizable and amenable for GPU computing, and its computational performance was substantially increased by the elimination of scattered interpolation. The accuracy and simulation time of the proposed method are compared against a finite element solution and also against experimental results from existing literature. The proposed method provides accuracy comparable to that of finite element methods, achieving an order of magnitude reduction in simulation time.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1142/S0218202520500141
