Two-phase flow problem plays an important role in many scientific and engineering processes. The problem can be described by a phase-field model consisting of the coupled Cahn-Hilliard and Navier-Stokes equations with...
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The work presents numerical results using adaptive BDDC deluxe methods for preconditioning the linear systems arising from finite element discretizations of the time-domain, quasi-static approximation of the Maxwell’...
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Algebraic flux correction (AFC) schemes are applied to the numerical solution of scalar steady-state convection- diffusion-reaction equations. A general result on the discrete maximum principle (DMP) is established un...
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ISBN:
(数字)9783319672021
ISBN:
(纸本)9783319672021;9783319672014
Algebraic flux correction (AFC) schemes are applied to the numerical solution of scalar steady-state convection- diffusion-reaction equations. A general result on the discrete maximum principle (DMP) is established under a weak assumption on the limiters and used for proving the DMP for a particular limiter of upwind type under an assumption that may hold also on non-Delaunay meshes. Moreover, a simple modification of this limiter is proposed that guarantees the validity of the DMP on arbitrary simplicial meshes. Furthermore, it is shown that AFC schemes do not provide sharp approximations of boundary layers if meshes do not respect the convection direction in an appropriate way.
Optimized Schwarz methods (OSM) are very popular methods which were introduced in Lions (On the Schwarz alternating method. III: a variant for nonoverlapping subdomains. In: Chan TF, Glowinski R, Périaux J, Widlu...
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The Lotka-Volterra (LV) system describes a simple predator-prey model in mathematical biology. The hungry Lotka-Volterra (hLV) system assumed that each predator preys on two or more species is an extension;those invol...
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ISBN:
(纸本)9783319624266;9783319624242
The Lotka-Volterra (LV) system describes a simple predator-prey model in mathematical biology. The hungry Lotka-Volterra (hLV) system assumed that each predator preys on two or more species is an extension;those involving summations and products of nonlinear terms are referred to as summation-type and product-type hLV systems, respectively. Time-discretizations of these systems are considered in the study of integrable systems, and have been shown to be applicable to computing eigenvalues of totally nonnegative (TN) matrices. Monotonic convergence to eigenvalues of TN matrices, however, has not yet been observed in the time-discretization of the product-type hLV system. In this paper, we show the existence of a center manifold associated with the time-discretization of the product-type hLV system, and then clarify how the solutions approach an equilibrium corresponding to the eigenvalues of TN matrices in the final phase of convergence.
The biharmonic equation is a fourth order equation, and thus needs two boundary conditions, and not just one like Laplace’s equation. While the clamped boundary condition can be taken naturally as the “Dirichlet” c...
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For a real symmetric-definite generalized eigenproblem of size N matrices Av = lambda Bv (B > 0), we solve those pairs whose eigenvalues are in a real interval [a, b] by the filter diagonalization method. In our pr...
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ISBN:
(数字)9783319624266
ISBN:
(纸本)9783319624266;9783319624242
For a real symmetric-definite generalized eigenproblem of size N matrices Av = lambda Bv (B > 0), we solve those pairs whose eigenvalues are in a real interval [a, b] by the filter diagonalization method. In our present study, the filter which we use is a real-part of a polynomial of a resolvent: F = Re Sigma(n )(k=1)gamma(k){R(rho)}(k). Here R(rho) = (A - rho B)(-1) B is the resolvent with a non-real complex shift rho, and gamma(k) are coefficients. In our experiments, the (half) degree n is 15 or 20. By tuning the shift rho and coefficients {gamma(k)} well, the filter passes those eigenvectors well whose eigenvalues are in a neighbor of [a, b], but strongly reduces those ones whose eigenvalues are separated from the interval. We apply the filter to a set of sufficiently many B-orthonormal random vectors {x((l))}to obtain another set {y((l))}. From both sets of vectors and properties of the filter, we construct a basis which approximately spans an invariant subspace whose eigenvalues are in a neighbor of [a, b]. An application of the Rayleigh-Ritz procedure to the basis gives approximations of all required eigenpairs. Experiments for banded problems showed this approach worked in success.
Elliptic problems with oscillating coefficients can be approximated up to arbitrary accuracy by using sufficiently fine meshes, i.e., by resolving the fine scale. Well-known multiscale finite elements (Henning et al. ...
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The Nagumo equation is a simple nonlinear reaction-diffusion equation, which has important applications in neuroscience and biological electricity. If the equation is reaction-dominated, numerical oscillationsmay appe...
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ISBN:
(数字)9783319672021
ISBN:
(纸本)9783319672021;9783319672014
The Nagumo equation is a simple nonlinear reaction-diffusion equation, which has important applications in neuroscience and biological electricity. If the equation is reaction-dominated, numerical oscillationsmay appear near the traveling wave front, which makes it challenging to find stable solutions. In the present study, a new method is developed on uniform meshes to solve the Nagumo equation. Numerical results are given to demonstrate the performance of the algorithm. Convergence rates with respect to spatial and temporal discretization are obtained experimentally. Some properties of the nerve model are confirmed numerically.
The Bethe-Salpeter eigenvalue problem is solved in condense matter physics to estimate the absorption spectrum of solids. It is a structured eigenvalue problem. Its special structure appears in other approaches for st...
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ISBN:
(数字)9783319624266
ISBN:
(纸本)9783319624266;9783319624242
The Bethe-Salpeter eigenvalue problem is solved in condense matter physics to estimate the absorption spectrum of solids. It is a structured eigenvalue problem. Its special structure appears in other approaches for studying electron excitation in molecules or solids also. When the Bethe-Salpeter Hamiltonian matrix is definite, the corresponding eigenvalue problem can be reduced to a symmetric eigenvalue problem. However, its special structure leads to a number of interesting spectral properties. We describe these properties that are crucial for developing efficient and reliable numerical algorithms for solving this class of problems.
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