Stochastic Advection-diffusion Equation Research Articles

A mixed-method to numerical simulation of variable order stochastic advection diffusion equations

The study of stochastic problems is very important and there is an increasing demand for investigating the behavior of a number of sophisticated dynamical systems in different areas of science as well as in engineering and finance. In this paper, we present a method which is based on the collocation method (CM) and the operational matrix method (OMM) by using Hosoya polynomials (HPs). This mixed-method helps us to convert the stochastic problems into a system of algebraic equations to access a solution of such problems. One of the most important advantages of this method is the simulation of the noise terms at the collocation points. Finally, two types of the VO–SADEs are simulated and tested via the proposed technique to confirm the superiority and accuracy.

A Reduced-Order Finite Difference Scheme Based on POD for Fractional Stochastic Advection–Diffusion Equation

A Reduced-Order Finite Difference Scheme Based on POD for Fractional Stochastic Advection–Diffusion Equation

Two-timescale stochastic gradient descent in continuous time with applications to joint online parameter estimation and optimal sensor placement

In this paper, we establish the almost sure convergence of two-timescale stochastic gradient descent algorithms in continuous time under general noise and stability conditions, extending well known results in discrete time. We analyse algorithms with additive noise and those with non-additive noise. In the non-additive case, our analysis is carried out under the assumption that the noise is a continuous-time Markov process, controlled by the algorithm states. The algorithms we consider can be applied to a broad class of bilevel optimisation problems. We study one such problem in detail, namely, the problem of joint online parameter estimation and optimal sensor placement for a partially observed diffusion process. We demonstrate how this can be formulated as a bilevel optimisation problem, and propose a solution in the form of a continuous-time, two-timescale, stochastic gradient descent algorithm. Furthermore, under suitable conditions on the latent signal, the filter, and the filter derivatives, we establish almost sure convergence of the online parameter estimates and optimal sensor placements to the stationary points of the asymptotic log-likelihood and asymptotic filter covariance, respectively. We also provide numerical examples, illustrating the application of the proposed methodology to a partially observed Beneš equation, and a partially observed stochastic advection-diffusion equation.

Approximation of SPDE covariance operators by finite elements: a semigroup approach

Abstract The problem of approximating the covariance operator of the mild solution to a linear stochastic partial differential equation is considered. An integral equation involving the semigroup of the mild solution is derived and a general error decomposition is proven. This formula is applied to approximations of the covariance operator of a stochastic advection-diffusion equation and a stochastic wave equation, both on bounded domains. The approximations are based on finite element discretizations in space and rational approximations of the exponential function in time. Convergence rates are derived in the trace class and Hilbert–Schmidt norms with numerical simulations illustrating the results.

Strongly Nonlinear Diffusion in Turbulent Environment: A Problem with Infinitely Many Couplings

The field theoretic renormalization group is applied to the strongly nonlinear stochastic advection-diffusion equation. The turbulent advection is modelled by the Kazantsev–Kraichnan “rapid-change” ensemble. As a requirement of the renormalizability, the model necessarily involves infinite number of coupling constants (“charges”). The one-loop counterterm is calculated explicitly. The corresponding renormalization group equation demonstrates existence of a pair of two-dimensional surfaces of fixed points in the infinite-dimensional parameter space. If the surfaces contain infrared attractive regions, the problem allows for the large-scale, long-time scaling behaviour. For the first surface (advection is irrelevant), the critical dimensions of the scalar field Δθ, the response field Δθ′ and the frequency Δω are nonuniversal (through the dependence on the effective couplings) but satisfy certain exact identities. For the second surface (advection is relevant), the dimensions are universal and they are found exactly.

Joint Online Parameter Estimation and Optimal Sensor Placement for the Partially Observed Stochastic Advection-Diffusion Equation

In this paper, we consider the problem of jointly performing online parameter estimation and optimal sensor placement for a partially observed infinite dimensional linear diffusion process. We present a novel solution to this problem in the form of a continuous-time, two-timescale stochastic gradient descent algorithm, which recursively seeks to maximise the log-likelihood with respect to the unknown model parameters, and to minimise the expected mean squared error of the hidden state estimate with respect to the sensor locations. We also provide extensive numerical results illustrating the performance of the proposed approach in the case that the hidden signal is governed by the two-dimensional stochastic advection-diffusion equation.

Finite Difference and Spline Approximation for Solving Fractional Stochastic Advection-Diffusion Equation

This paper is concerned with numerical solution of time fractional stochastic advection-diffusion type equation where the first order derivative is substituted by a Caputo fractional derivative of order $$\alpha $$ ( $$0 <\alpha \le 1$$ ). This type of equations due to randomness can rarely be solved, exactly. In this paper, a new approach based on finite difference method and spline approximation is employed to solve time fractional stochastic advection-diffusion type equation, numerically. After implementation of proposed method, the under consideration equation is transformed to a system of second order differential equations with appropriate boundary conditions. Then, using a suitable numerical method such as the backward differentiation formula, the resulting system can be solved. In addition, the error analysis is shown in some mild conditions by ignoring the error terms $$O(\Delta t^2)$$ in the system. In order to show the pertinent features of the suggested algorithm such as accuracy, efficiency and reliability, some test problems are included. Comparison achieved results via proposed scheme in the case of classical stochastic advection-diffusion equation ( $$\alpha =1$$ ) with obtained results via wavelets Galerkin method and obtained results for other values of $$\alpha $$ with the values of exact solution confirm the validity, efficiency and applicability of the proposed method.

Hybrid Hermite polynomial chaos SBP-SAT technique for stochastic advection-diffusion equations

The study of advection–diffusion equation has relatively became an active research topic in the field of uncertainty quantification (UQ) due to its numerous real life applications. In this paper, Hermite polynomial chaos is united with summation-by-parts (SBP) – simultaneous approximation terms (SATs) technique to solve the advection–diffusion equations with random Dirichlet boundary conditions (BCs). Polynomial chaos expansion (PCE) with Hermite basis is employed to separate the randomness, then SBP operators are used to approximate the differential operators and SATs are used to enforce BCs by ensuring the stability. For each chaos coefficient, time integration is performed using Runge–Kutta method of fourth order (RK4). Statistical moments namely mean and variance are computed using polynomial chaos coefficients without any extra computational effort. The method is applied on three test problems for validation. The first two test problems are stochastic advection equations on [Formula: see text] without any boundary and third problem is stochastic advection–diffusion equation on [0,2] with Dirichlet BCs. In case of third problem, we have obtained a range of permissible parameters for a stable numerical solution.

A numerical scheme for solutions of stochastic advection-diffusion equations

In this paper, finite difference schemes are proposed to approximate solutions of stochastic advection-diffusion equations. We used central-difference formula of third-order to approximate spatial derivatives. The stability, consistency and convergence of the scheme are analysed and established. A numerical result is also given to demonstrate the computational efficiency of the stochastic schemes.

Combination of finite difference method and meshless method based on radial basis functions to solve fractional stochastic advection–diffusion equations

The present article develops a semi-discrete numerical scheme to solve the time-fractional stochastic advection–diffusion equations. This method, which is based on finite difference scheme and radial basis functions (RBFs) interpolation, is applied to convert the solution of time-fractional stochastic advection–diffusion equations to the solution of a linear system of algebraic equations. The mechanism of this method is such that time-fractional stochastic advection–diffusion equation is first transformed into elliptic stochastic differential equations by using finite difference scheme. Then meshfree method based on RBFs has been used to approximate the resulting equation. In other words, the approximate solution of time-fractional stochastic advection–diffusion equation is achieved with discrete the domain in the t-direction by finite difference method and approximating the unknown function in the x-direction by generalized inverse multiquadrics RBFs. In this method, the noise terms are directly simulated at the collocation points in each time step and it is the most important advantage of the suggested approach. Stability and convergence of the scheme are established. Finally, some test problems are included to confirm the accuracy and efficiency of the new approach.

Stochastic variational multiscale analysis of the advection–diffusion equation: Advective–diffusive regime and multi-dimensional problems

We present the variational multiscale (VMS) formulation for the stochastic advection–diffusion equation in which the source of uncertainty may arise from random advective and diffusive fields (or material parameters), a random source term, or a combination among them. In this formulation, a stabilization parameter arises that, for the advection–diffusion equation, is a complicated function of the uncertain advection and diffusion parameters. To efficiently incorporate the stochastic stabilization parameter in the numerical method, we develop a projection-based approach to obtain an approximation of the stabilization parameter in the form of a generalized polynomial chaos (gPC) expansion. We demonstrate the current approach for problems which span the advective and diffusive regimes in the stochastic domain. Specifically, four cases are considered with multi-dimensional physical and stochastic domains. These cases include an advection-dominated case with an uncertain advection parameter, a case spanning both the advective and diffusive regimes with uncertain advection and diffusion parameters, a case with an uncertain source term, and a case with scalar transport under an uncertain advection field in a channel with multiple branches. For the latter case uniform and non-uniform meshes are employed. The expectation and variance from the stochastic Galerkin and VMS formulations are compared against their counterparts from analytical or Monte Carlo sampling methods where appropriate. In summary, an accurate and stable numerical solution is obtained using the VMS formulation with projection-based approximation of the stochastic stabilization parameter for multi-dimensional problems including non-trivial physical discretization.

Logarithmic violation of scaling in strongly anisotropic turbulent transfer of a passive vector field.

Inertial-range asymptotic behavior of a vector (e.g., magnetic) field, passively advected by a strongly anisotropic turbulent flow, is studied by means of the field-theoretic renormalization group and the operator product expansion. The advecting velocity field is Gaussian, not correlated in time, with the pair correlation function of the form ∝δ(t-t')/k(⊥)(d-1+ξ), where k(⊥)=|k(⊥)| and k(⊥) is the component of the wave vector, perpendicular to the distinguished direction ("direction of the flow")--the d-dimensional generalization of the ensemble introduced by Avellaneda and Majda [Commun. Math. Phys. 131, 381 (1990)]. The stochastic advection-diffusion equation for the transverse (divergence-free) vector field includes, as special cases, the kinematic dynamo model for magnetohydrodynamic turbulence and the linearized Navier-Stokes equation. In contrast to the well-known isotropic Kraichnan's model, where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents, here the dependence on the integral turbulence scale L has a logarithmic behavior: Instead of powerlike corrections to ordinary scaling, determined by naive (canonical) dimensions, the anomalies manifest themselves as polynomials of logarithms of L. The key point is that the matrices of scaling dimensions of the relevant families of composite operators appear nilpotent and cannot be diagonalized. The detailed proof of this fact is given for the correlation functions of arbitrary order.

Meshless simulation of stochastic advection–diffusion equations based on radial basis functions

In this paper, a numerical technique is proposed for solving the stochastic advection–diffusion equations. Firstly, using the finite difference scheme, we transform the stochastic advection–diffusion equations into elliptic stochastic partial differential equations (SPDEs). Then the method of radial basis functions (RBFs) based on pseudospectral (PS) approach has been used to approximate the resulting elliptic SPDEs. In this study, we have used generalized inverse multiquadrics (GIMQ) RBFs, to approximate functions in the presented method. The main advantage of the proposed method over traditional numerical approaches is directly simulating the noise terms at the collocation points in each time step. To confirm the accuracy of the new approach and to show the performance of the selected RBFs, four examples are presented in one, two and three dimensions in regular and irregular domains. For test problems the statistical moments such as mean, variance and standard deviation are computed.

Smoothed particle hydrodynamics model for Landau-Lifshitz-Navier-Stokes and advection-diffusion equations.

We propose a novel smoothed particle hydrodynamics (SPH) discretization of the fully coupled Landau-Lifshitz-Navier-Stokes (LLNS) and stochastic advection-diffusion equations. The accuracy of the SPH solution of the LLNS equations is demonstrated by comparing the scaling of velocity variance and the self-diffusion coefficient with kinetic temperature and particle mass obtained from the SPH simulations and analytical solutions. The spatial covariance of pressure and velocity fluctuations is found to be in a good agreement with theoretical models. To validate the accuracy of the SPH method for coupled LLNS and advection-diffusion equations, we simulate the interface between two miscible fluids. We study formation of the so-called "giant fluctuations" of the front between light and heavy fluids with and without gravity, where the light fluid lies on the top of the heavy fluid. We find that the power spectra of the simulated concentration field are in good agreement with the experiments and analytical solutions. In the absence of gravity, the power spectra decay as the power -4 of the wavenumber-except for small wavenumbers that diverge from this power law behavior due to the effect of finite domain size. Gravity suppresses the fluctuations, resulting in much weaker dependence of the power spectra on the wavenumber. Finally, the model is used to study the effect of thermal fluctuation on the Rayleigh-Taylor instability, an unstable dynamics of the front between a heavy fluid overlaying a light fluid. The front dynamics is shown to agree well with the analytical solutions.

A Recursive Sparse Grid Collocation Method for Differential Equations with White Noise

We consider a sparse grid collocation method in conjunction with a time discretization of the differential equations for computing expectations of functionals of solutions to differential equations perturbed by time-dependent white noise. We first analyze the error of Smolyak's sparse grid collocation used to evaluate expectations of functionals of solutions to stochastic differential equations discretized by the Euler scheme. We show theoretically and numerically that this algorithm can have satisfactory accuracy for small magnitude of noise or small integration time, however it does not converge neither with decrease of the Euler scheme's time step size nor with increase of Smolyak's sparse grid level. Subsequently, we use this method as a building block for proposing a new algorithm by combining sparse grid collocation with a recursive procedure. This approach allows us to numerically integrate linear stochastic partial differential equations over longer times, which is illustrated in numerical tests on a stochastic advection-diffusion equation.

APPROXIMATION OF STOCHASTIC ADVECTION-DIFFUSION EQUATION USING COMPACT FINITE DIFFERENCE TECHNIQUE

In this paper, we propose a new method for solving the stochastic advection-diffusion equation of Ito type. In this work, we use a compact finite difference approximation for discretizing spatial derivatives of the mentioned equation and semi-implicit Milstein scheme for the resulting linear stochastic system of differential equation. The main purpose of this paper is the stability investigation of the applied method. Finally, some numerical examples are provided to show the accuracy and efficiency of the proposed technique.

Approximation of stochastic advection diffusion equations with Stochastic Alternating Direction Explicit methods

The numerical solutions of stochastic partial differential equations of Itô type with time white noise process, using stable stochastic explicit finite difference methods are considered in the paper. Basically, Stochastic Alternating Direction Explicit (SADE) finite difference schemes for solving stochastic time dependent advection-diffusion and diffusion equations are represented and the main properties of these stochastic numerical methods, e.g. stability, consistency and convergence are analyzed. In particular, it is proved that when stable alternating direction explicit schemes for solving linear parabolic PDEs are developed to the stochastic case, they retain their unconditional stability properties applying to stochastic advection-diffusion and diffusion SPDEs. Numerically, unconditional stable SADE techniques are significant for approximating the solutions of the proposed SPDEs because they do not impose any restrictions for refining the computational domains. The performance of the proposed methods is tested for stochastic diffusion and advection-diffusion problems, and the accuracy and efficiency of the numerical methods are demonstrated.

A Comparison of Closures for Stochastic Advection-Diffusion Equations

Perturbation-based moment equations for advection-diffusion equations with random advection have been shown to produce physically unrealistic multimodality. Despite similarities, macrodispersion theory applied to the same equations produces moments that do not exhibit such multimodal behavior. This is because macrodispersion approximations, whether explicitly or implicitly, involve renormalized perturbations that remove secularity by including select higher-order terms. We consider basic differences between the two approaches, using a low-order macrodispersion approximation to clarify why one produces physically meaningful behavior while the other does not. We demonstrate that using a conventional asymptotic expansion (in the order of velocity fluctuations) leads to equations that cannot produce physically meaningful (macro)dispersion, whether applied to moment equations or macrodispersion theory, and prove that the resulting moment equations are in fact hyperbolic for the special case of pure advection b...

Langevin model for reactive transport in porous media

Existing continuum models for reactive transport in porous media tend to overestimate the extent of solute mixing and mixing-controlled reactions because the continuum models treat both the mechanical and diffusive mixings as an effective Fickian process. Recently, we have proposed a phenomenological Langevin model for flow and transport in porous media [A. M. Tartakovsky, D. M. Tartakovsky, and P. Meakin, Phys. Rev. Lett. 101, 044502 (2008)]. In the Langevin model, the fluid flow in a porous continuum is governed by a combination of a Langevin equation and a continuity equation. Pore-scale velocity fluctuations, the source of mechanical dispersion, are represented by the white noise. The advective velocity (the solution of the Langevin flow equation) causes the mechanical dispersion of a solute. Molecular diffusion and sub-pore-scale Taylor-type dispersion are modeled by an effective stochastic advection-diffusion equation. Here, we propose a method for parameterization of the model for a synthetic porous medium, and we use the model to simulate multicomponent reactive transport in the porous medium. The detailed comparison of the results of the Langevin model with pore-scale and continuum (Darcy) simulations shows that: (1) for a wide range of Peclet numbers the Langevin model predicts the mass of reaction product more accurately than the Darcy model; (2) for small Peclet numbers predictions of both the Langevin and the Darcy models agree well with a prediction of the pore-scale model; and (3) the accuracy of the Langevin and Darcy model deteriorates with the increasing Peclet number but the accuracy of the Langevin model decreases more slowly than the accuracy of the Darcy model. These results show that the separate treatment of advective and diffusive mixing in the stochastic transport model is more accurate than the classical advection-dispersion theory, which uses a single effective diffusion coefficient (the dispersion coefficient) to describe both types of mixing.

Turbulence with pressure: anomalous scaling of a passive vector field.

The field theoretic renormalization group (RG) and the operator-product expansion are applied to the model of a transverse (divergence-free) vector quantity, passively advected by the "synthetic" turbulent flow with a finite (and not small) correlation time. The vector field is described by the stochastic advection-diffusion equation with the most general form of the inertial nonlinearity; it contains as special cases the kinematic dynamo model, linearized Navier-Stokes (NS) equation, the special model without the stretching term that possesses additional symmetries and has a close formal resemblance with the stochastic NS equation. The statistics of the advecting velocity field is Gaussian, with the energy spectrum E(k) proportional to k(1-epsilon) and the dispersion law omega proportional to k(-2+eta), k being the momentum (wave number). The inertial-range behavior of the model is described by seven regimes (or universality classes) that correspond to nontrivial fixed points of the RG equations and exhibit anomalous scaling. The corresponding anomalous exponents are associated with the critical dimensions of tensor composite operators built solely of the passive vector field, which allows one to construct a regular perturbation expansion in epsilon and eta; the actual calculation is performed to the first order (one-loop approximation), including the anisotropic sectors. Universality of the exponents, their (in)dependence on the forcing, effects of the large-scale anisotropy, compressibility, and pressure are discussed. In particular, for all the scaling regimes the exponents obey a hierarchy related to the degree of anisotropy: the more anisotropic is the contribution of a composite operator to a correlation function, the faster it decays in the inertial range. The relevance of these results for the real developed turbulence described by the stochastic NS equation is discussed.