An EMC-HDG scheme for the convection-diffusion equation with random diffusivity

Numerical Algorithms - In this work, we propose an ensemble Monte Carlo hybridizable discontinuous Galerkin (EMC-HDG) algorithm to simulate the convection-diffusion equation with random diffusion...     

Normal approximation for a random elliptic equation

We consider solutions of an elliptic partial differential equation in \(\mathbb{R }^d\) with a stationary, random conductivity coefficient that is also periodic with period \(L\) . Boundary conditions on a square domain of width \(L\) are arranged so that the solution has a macroscopic unit gradient. We then consider the average flux that results from this imposed boundary condition. It is known that in the limit \(L \rightarrow \infty \) , this quantity converges to a deterministic constant, almost surely. Our main result is that the law of this random variable is very close to that of a normal random variable, if the domain size \(L\) is large. We quantify this approximation by an error estimate in total variation. The error estimate relies on a second order Poincaré inequality developed recently by Chatterjee.     

Diffusion approximation of controlled branching processes with random environments

Conditions are established under which a sequence of density dependent branching processes with random environments converges weakly to a diffusion process. The limiting diffusion process can be obtained as a solution of a stochastic differential equation     

Finite element algorithm based on high-order time approximation for time fractional convection-diffusion equation

In this paper, finite element method with high-order approximation for time fractional derivative is considered and discussed to find the numerical solution of time fractional convection-diffusion equation. Some lemmas are introduced and proved, further the stability and error estimates are discussed and analyzed, respectively. The convergence result $O(h^{r+1}+\tau^{3-\alpha})$ can be derived, which illustrates that time convergence rate is higher than the order $(2-\alpha)$ derived by $L1$-approximation. Finally, to validate our theoretical results, some computing data are provided.     

Uniform grid approximation of nonsmooth solutions with improved convergence for a singularly perturbed convection-diffusion equation with characteristic layers

A mixed boundary value problem for a singularly perturbed elliptic convection-diffusion equation with constant coefficients in a square domain is considered. Dirichlet conditions are specified on two sides orthogonal to the flow, and Neumann conditions are set on the other two sides. The right-hand side and the boundary functions are assumed to be sufficiently smooth, which ensures the required smoothness of the desired solution in the domain, except for neighborhoods of the corner points. Only zero-order compatibility conditions are assumed to hold at the corner points. The problem is solved numerically by applying an inhomogeneous monotone difference scheme on a rectangular piecewise uniform Shishkin mesh. The inhomogeneity of the scheme lies in that the approximating difference equations are not identical at different grid nodes but depend on the perturbation parameter. Under the assumptions made, the numerical solution is proved to converge ?-uniformly to the exact solution in a discrete uniform metric at an O(N ^?3/2ln² N) rate, where N is the number of grid nodes in each coordinate direction.     

A semi-circulant preconditioner for the convection-diffusion equation

In this paper we define and analyze a semi-circulant preconditioner for the convection-diffusion equation. We derive analytical formulas for the eigenvalues and the eigenvectors of the preconditioned system of equations. We show that for mesh Péclet numbers less than 2, the rate of convergence depends only on the mesh Péclet number and the direction of the convective field and not on the spatial grid ratio or the number of unknowns. Received February 20, 1997 / Revised version received November 19, 1997     

Galerkin-petrov limit schemes for the convection-diffusion equation

In the present paper, we suggest a method for constructing grid schemes for the multidimensional convection-diffusion equation. The method is based on the approximation of the integral identity that is used in the definition of a weak solution of the differential problem. The use of spaces of smooth trial functions and spaces of functions with possible discontinuities in which the solution of the original problem is sought naturally leads to Galerkin-Petrov methods. The suggested method for the construction of grid schemes is based on a finite-element semidiscretization of the original space with respect to space variables, which constructs the space of trial functions on the basis of the direction of the convective transport near the boundaries of finite elements, the limit passage from a scheme with smooth trial functions to schemes with discontinuous trial functions, and the further discretization of the resulting equations with respect to the time variable. We prove the stability of the constructed difference schemes and present the results of computations for model problems.     

Random collision model for random genetic drift and stochastic difference equation

Summary At first we introduce a simple stochastic difference equation, to simulate random sampling drift in population genetics, which is naturally obtained from a random collision model. Next, we introduce a random collision model to simulate overdominance model in population genetics. We assume in a time interval °t, a random collision of four particles, which represents overdominant selection, takes place at a certain probability, where a particle corresponds to a gene. We assume that mutation takes place by some rate and assume that every new mutation is different from extant alleles. We estimate mean heterozygosity by our simulation method and compare it with the result obtained by using a stochastic difference equation for overdominance model. The Institute of Statistical Mathematics     

非线性对流扩散方程的守恒律

利用直接方法研究了非线性对流扩散方程的守恒律,得到了关于非线性对流扩散方程的守恒律乘子性质的一个定理.利用这个定理,可以简化守恒律乘子的确定方程.随后通过对确定方程中的变量函数进行分析,发现在四种情况下乘子的确定方程是可解的.最后解出这些守恒律乘子,利用积分公式法分别得到了四种情况下对应于各个守恒律乘子的守恒律.     

A nonlocal convection-diffusion equation

In this paper we study a nonlocal equation that takes into account convective and diffusive effects, ut=J∗u−u+G∗(f(u))−f(u) in Rd, with J radially symmetric and G not necessarily symmetric. First, we prove existence, uniqueness and continuous dependence with respect to the initial condition of solutions. This problem is the nonlocal analogous to the usual local convection-diffusion equation ut=Δu+b⋅∇(f(u)). In fact, we prove that solutions of the nonlocal equation converge to the solution of the usual convection-diffusion equation when we rescale the convolution kernels J and G appropriately. Finally we study the asymptotic behaviour of solutions as t→∞ when f(u)=|u|^q−1u with q>1. We find the decay rate and the first-order term in the asymptotic regime.     

Self-similar solutions for a convection-diffusion equation with absorption inR N

We prove the existence of a positive and smooth solution for the following semilinear elliptic problem: % MathType!End!2!1! for anya∈R ^N , 1<p<1+2/N andq=(p+1)/2. This solution decays exponentially as |x|→+∞. Moreover, if |a| is sufficiently small, this positive and rapidly decaying solution is unique. The existence of a positive, self-similar solution % MathType!End!2!1! follows for the following convection-diffusion equation with absorption: % MathType!End!2!1!. It is also a very singular solution. This solution decays as |x|→+∞ for anyt>0 fixed. Because of the nonvariational nature of the elliptic problem, a fixed point method is used for proving the existence result. The uniqueness is proved applying the Implicit Function Theorem. The work of the first author has been partially supported by Grant 1273/00003/88 of the University of the Basque Country. The work of the second author has been supported by Grant PB 86-0112-C02-00 of the Dirección General de Investigación Científica y Técnica.     

Weak approximation of CIR equation by discrete random variables

For the CIR equation \( {\text{d}}{X_t} = \left( {\theta - k\,{X_t}} \right){\text{d}}t + \sigma \sqrt {{{X_t}}} {\text{d}}{B_t} \), we propose positive weak first- and second-order approximations that use, at each step, generation of discrete (respectively two- and three-valued) random variables (Theorems 3 and 4). The equation is split into deterministic part \( {\text{d}}{D_t} = \left( {\theta - k{D_t}} \right){\text{d}}t \), which is solved exactly, and stochastic part \( {\text{d}}{S_t} = \sigma \sqrt {{{S_t}}} {\text{d}}{B_t} \), which is actually approximated in distribution.     

Diffusion approximation for the components in critical inhomogeneous random graphs of rank 1.

Consider the random graph on n vertices 1,…,n. Each vertex i is assigned a type x_i with x₁,…,x_n being independent identically distributed as a nonnegative random variable X. We assume that EX³< ∞. Given types of all vertices, an edge exists between vertices i and j independent of anything else and with probability \begin{align*}\min \{1, \frac{x_ix_j}{n}\left(1+\frac{a}{n^{1/3}} \right) \}\end{align*}. We study the critical phase, which is known to take place when EX² = 1. We prove that normalized by n^‐2/3the asymptotic joint distributions of component sizes of the graph equals the joint distribution of the excursions of a reflecting Brownian motion with diffusion coefficient \begin{align*}\sqrt{{\textbf{ E}}X{\textbf{ E}}X^3}\end{align*}and drift \begin{align*}a-\frac{{\textbf{ E}}X^3}{{\textbf{ E}}X}s\end{align*}. In particular, we conclude that the size of the largest connected component is of order n^2/3. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 43, 486–539, 2013     

Discretization of a convection-diffusion equation

The unsteady convection-diffusion equation with constant coefficientsadmits an exact solution in the form of a convolution integral,which provides an explicit representation of the evolution operatorthrough one time step. This is used to unify many numericalschemes for the equation, showing interrelationships betweenfinite difference and finite element schemes and presentinga general framework for detailed error analysis. In particular,the upwind scheme, Lax Wendroff, QUICKEST, the ECG schemes andCrank Nicolson are all members of a family that includes powerfulnew schemes. Fourier analysis is used to obtain practical stabilityregions and some insights into accuracy; and the Peano kerneltheorem is also used to derive rigorous error bounds that canbe generalized to irregular meshes.     

Uniform mesh approximation to nonsmooth solutions of a singularly perturbed convection-diffusion equation in a rectangle

Here nonsmooth solutions of a differential equation are treated as solutions for which the compatibility conditions are not required to hold at the corner points of the domain and hence corner singularities can occur. In the present paper, we drop the compatibility conditions at three of the four vertices of a rectangle. At the remaining vertex, from which a characteristic (inclined) of the reduced equation issues, we impose compatibility conditions providing the C ^3,λ -smoothness of the desired solution in a neighborhood of that vertex as well as additional conditions leading to the smoothness of solutions of the reduced equation occurring in the regular component of the solution of the considered problem. Under our assumptions and for a sufficient smoothness of the coefficients of the equation and its right-hand side, we show that the classical five-point upwind approximation on a Shishkin piecewise uniform mesh preserves the accuracy specific for the smooth case; i.e., the mesh solution uniformly (with respect to a small parameter) converges in the L _∞ ^h -norm to the exact solution at the rate O(N ⁻¹ ln² N), where N is the number of mesh nodes in each of the coordinate directions.     

Limit Galerkin-Petrov schemes for a nonlinear convection-diffusion equation

In the present paper, we suggest a version of the nonconformal finite-element method (a perturbed Galerkin method) for approximating a quasilinear convection-diffusion equation in divergence form. A grid scheme is constructed with the use of an approach based on the Galerkin-Petrov approximation to the mixed statement of the original problem. The separated coordinate approximation of the solution components for the mixed problem permits one to take into account the direction of convective transport and preserve the main properties of the spatial operator of the original problem. We prove the stability of the line method scheme and a two-layer weighted scheme for the original problem.     

The convection-diffusion equation for a finite domain with time varying boundaries

A solution is developed for a convection-diffusion equation describing chemical transport with sorption, decay, and production. The problem is formulated in a finite domain where the appropriate conservation law yields Robin conditions at the ends. When the input concentration is arbitrary, the problem is underdetermined because of an unknown exit concentration. We resolve this by defining the exit concentration as a solution to a similar diffusion equation which satisfies a Dirichlet condition at the left end of the half line. This problem does not appear to have been solved in the literature, and the resulting representation should be useful for problems of practical interest.

Authors of previous works on problems of this type have eliminated the unknown exit concentration by assuming a continuous concentration at the outflow boundary. This yields a well-posed problem by forcing a homogeneous Neumann exit, widely known as Danckwerts condition. We provide a solution to that problem and use it to produce an estimate which demonstrates that Danckwerts condition implies a zero concentration at the outflow boundary, even for a long flow domain and a large time.     

Convergence of high-order deterministic particle methods for the convection-diffusion equation

A proof of high-order convergence of three deterministic particle methods for the convection-diffusion equation in two dimensions is presented. The methods are based on discretizations of an integro-differential equation in which an integral operator approximates the diffusion operator. The methods differ in the discretization of this operator. The conditions for convergence imposed on the kernel that defines the integral operator include moment conditions and a condition on the kernel's Fourier transform. Explicit formulae for kernels that satisfy these conditions to arbitrary order are presented. © 1997 John Wiley & Sons, Inc.     

Ergodicity for the Navier‐Stokes equation with degenerate random forcing: Finite‐dimensional approximation

We study Galerkin truncations of the two‐dimensional Navier‐Stokes equation under degenerate, large‐scale, stochastic forcing. We identify the minimal set of modes that has to be forced in order for the system to be ergodic. Our results rely heavily on the structure of the nonlinearity. © 2001 John Wiley & Sons, Inc.