Linear Convection-diffusion Equation Research Articles

EXtended Hybridizable Discontinuous Galerkin (X-HDG) method for linear convection-diffusion equations on unfitted domains

In this work, we propose a novel strategy for the numerical solution of linear convection diffusion equation (CDE) over unfitted domains. In the proposed numerical scheme, strategies from high order Hybridized Discontinuous Galerkin method and eXtended Finite Element method are combined with the level set definition of the boundaries. The proposed scheme and hence, is named as eXtended Hybridizable Discontinuous Galerkin (XHDG) method. In this regard, the Hybridizable Discontinuous Galerkin (HDG) method is eXtended to the unfitted domains; i.e., the computational mesh does not need to fit to the domain boundary; instead, the boundary is defined by a level set function and cuts through the background mesh arbitrarily. The original unknown structure of HDG and its hybrid nature ensuring the local conservation of fluxes is kept, while developing a modified bilinear form for the elements cut by the boundary. At every cut element, an auxiliary nodal trace variable on the boundary is introduced, which is eliminated afterwards while imposing the boundary conditions. Both stationary and time dependent CDEs are studied over a range of flow regimes from diffusion to convection dominated; using high order (p≤4) XHDG through benchmark numerical examples over arbitrary unfitted domains. Results proved that XHDG inherits optimal (p+1) and super (p+2) convergence properties of HDG while removing the fitting mesh restriction.

Uniform stability for local discontinuous Galerkin methods with implicit-explicit Runge-Kutta time discretizations for linear convection-diffusion equation

In this paper, we consider the linear convection-diffusion equation in one dimension with periodic boundary conditions, and analyze the stability of fully discrete methods that are defined with local discontinuous Galerkin (LDG) methods in space and several implicit-explicit (IMEX) Runge-Kutta methods in time. By using the forward temporal differences and backward temporal differences, respectively, we establish two general frameworks of the energy-method based stability analysis. From here, the fully discrete schemes being considered are shown to have monotonicity stability, i.e. the L 2 L^2 norm of the numerical solution does not increase in time, under the time step condition τ ≤ F ( h / c , d / c 2 ) \tau \le \mathcal {F}(h/c, d/c^2) , with the convection coefficient c c , the diffusion coefficient d d , and the mesh size h h . The function F \mathcal {F} depends on the specific IMEX temporal method, the polynomial degree k k of the discrete space, and the mesh regularity parameter. Moreover, the time step condition becomes τ ≲ h / c \tau \lesssim h/c in the convection-dominated regime and it becomes τ ≲ d / c 2 \tau \lesssim d/c^2 in the diffusion-dominated regime. The result is improved for a first order IMEX-LDG method. To complement the theoretical analysis, numerical experiments are further carried out, leading to slightly stricter time step conditions that can be used by practitioners. Uniform stability with respect to the strength of the convection and diffusion effects can especially be relevant to guide the choice of time step sizes in practice, e.g. when the convection-diffusion equations are convection-dominated in some sub-regions.

On the uniqueness of linear convection–diffusion equations with integral boundary conditions

We investigate a class of convection–diffusion equations in an expanding domain involving a parameter, where we consider integral boundary conditions that depend non-locally on unknown solutions. Generally, the uniqueness result of this type of equation is unclear. In this work, we obtain a uniqueness result when the domain is sufficiently large or small. This approach has the advantage of transforming the integral boundary conditions into new Dirichlet boundary conditions so that we can obtain refined estimates, and the comparison theorem can be applied to the equations. Furthermore, we show a domain such that under different boundary data, the equation in this domain can have infinitely numerous solutions or no solution. This work may contribute to the first understanding of the domain size’s effect on the existence and uniqueness of the linear convection–diffusion equation with integral-type boundary conditions.

Operator-splitting local discontinuous Galerkin method for multi-dimensional linear convection-diffusion equations

Operator-splitting local discontinuous Galerkin method for multi-dimensional linear convection-diffusion equations

Novel Numerical Method Based on the Analog Equation Method for a Class of Anisotropic Convection-Diffusion Problems

In this work, a CMFS method based on the analogy equation method, the radial basis function and the method of fundamental solutions for linear and nonlinear convection-diffusion equations in anisotropic materials is presented. The analog equation method is utilized to transform the linear and nonlinear convection-diffusion equation into an equivalent one. The expressions of the homogeneous solution and particular solution are derived by utilizing the radial basis function approximation and the method of fundamental solutions, respectively. By enforcing the desired solution to satisfy the original convection-diffusion equation with boundary conditions at boundary and internal collocation points yield a nonlinear system of equations, which can be solved by using the Newton-Raphson iteration or the Picard method of iteration. The error convergence curves of the proposed meshless method have been investigated by using different globally supported radial basis functions. Numerical experiments show that the proposed CMFS method is promising for anisotropic convection-diffusion problems with accurate and stable results.

Analysis of Pseudo-spectral Methods Used for Numerical Simulations of Turbulence

Events in turbulent flows computed by direct numerical simulation (DNS) are often calibrated with properties based on homogeneous isotropic turbulence, advanced by Kolmogorov, and given in Turbulence, U. Frisch, Cambridge Univ. Press, UK (1995). However, these computational procedures are not calibrated using numerical analyses in order to assess their strengths and weaknesses for DNS. This is with the exception in "A critical assessment of simulations for transitional and turbulence flows- Sengupta, T.K., In Proc. of IUTAM Symp. on Advances in Computation, Modeling and Control of Transitional and Turbulent Flows, 491-532, (eds. Sengupta, Lele, Sreenivasan, Davidson), WSPC, Singapore (2016)", where such a calibration has been advocated for numerical schemes using global spectral analysis (GSA) for the convection equation. In recent times, due to growing computing power, simulations have been reported using pseudo-spectral methods, with spatial discretization performed in Fourier spectral space and time-integration by multi-stage Runge-Kutta (RK) methods. Here, we perform GSA of Fourier spectral methods for the first time with RK2 and other multistage Runge-Kutta methods using the model linear convection and convection-diffusion equations. With the help of GSA, various sources of numerical errors are quantified. The major limitations of the RK2-Fourier spectral method are demonstrated for DNS and alternate choices are presented. We specify optimal parameters to achieve the best possible accuracy for simulations. There is a one-to-one correspondence of numerical solutions obtained by linear convection-diffusion equation and nonlinear Navier-Stokes equation with respect to numerical parameters. This enables us to investigate the capabilities of the numerical methods for DNS.

Arbitrary Lagrangian–Eulerian Local Discontinuous Galerkin Method for Linear Convection–Diffusion Equations

Arbitrary Lagrangian–Eulerian Local Discontinuous Galerkin Method for Linear Convection–Diffusion Equations

On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows

The kinetic energy of a flow is proportional to the square of the L2(Ω) norm of the velocity. Given a sufficient regular velocity field and a velocity finite element space with polynomials of degree r, then the best approximation error in L2(Ω) is of order r+1. In this survey, the available finite element error analysis for the velocity error in L∞(0,T;L2(Ω)) is reviewed, where T is a final time. Since in practice the case of small viscosity coefficients or dominant convection is of particular interest, which may result in turbulent flows, robust error estimates are considered, i.e., estimates where the constant in the error bound does not depend on inverse powers of the viscosity coefficient. Methods for which robust estimates can be derived enable stable flow simulations for small viscosity coefficients on comparatively coarse grids, which is often the situation encountered in practice. To introduce stabilization techniques for the convection-dominated regime and tools used in the error analysis, evolutionary linear convection–diffusion equations are studied at the beginning. The main part of this survey considers robust finite element methods for the incompressible Navier–Stokes equations of order r−1, r, and r+1/2 for the velocity error in L∞(0,T;L2(Ω)). All these methods are discussed in detail. In particular, a sketch of the proof for the error bound is given that explains the estimate of important terms which determine finally the order of convergence. Among them, there are methods for inf–sup stable pairs of finite element spaces as well as for pressure-stabilized discretizations. Numerical studies support the analytic results for several of these methods. In addition, methods are surveyed that behave in a robust way but for which only a non-robust error analysis is available. The conclusion of this survey is that the problem of whether or not there is a robust method with optimal convergence order for the kinetic energy is still open.

Learning Partial Differential Equations from Noisy Data using Neural Networks

The problem of learning partial differential equations (PDEs) from given data is investigated here. Several algorithms have been developed for PDE learning from accurate data sets. These include using sparse optimization for approximating the coefficients of candidate terms in a general PDE model. In this work, the study is extended to spatiotemporal data sets with various noise levels. We compare the performance of conventional and novel methods for denoising the data. Different architectures of neural networks are used to denoise data and approximate derivatives. These methods are numerically tested on the linear convection-diffusion equation and the nonlinear convection-diffusion equation (Burgers’ equation). Results suggest that modification in the hidden units/hidden layers in the network architecture have an effect on the accuracy of approximations with a significant rate. This is a further improvement on the previously known denoising methods of finite differences, polynomial regression splines and single layer neural network.

High-resolution compact numerical method for the system of 2D quasi-linear elliptic boundary value problems and the solution of normal derivatives on an irrational domain with engineering applications

In this paper, we present a novel approach to attain fourth-order approximate solution of 2D quasi-linear elliptic partial differential equation on an irrational domain. In this approach, we use nine grid points with dissimilar mesh in a single compact cell. We also discuss appropriate fourth-order numerical methods for the solution of the normal derivatives on a dissimilar mesh. The method has been protracted for solving system of quasi-linear elliptic equations. The convergence analysis is discussed to authenticate the proposed numerical approximation. On engineering applications, we solve various test problems, such as linear convection–diffusion equation, Burgers’equation, Poisson equation in singular form, NS equations, bi- and tri-harmonic equations and quasi-linear elliptic equations to show the efficiency and accuracy of the proposed methods. A comprehensive comparative computational experiment shows the accuracy, reliability and credibility of the proposed computational approach.

Stabilization and a posteriori error analysis of a mixed FEM for convection–diffusion problems with mixed boundary conditions

We introduce a new augmented dual-mixed finite element method for the linear convection-diffusion equation with mixed boundary conditions. The approach is based on adding suitable residual type terms to a dual-mixed formulation of the problem. We prove that for appropriate values of the stabilization parameters, that depend on the diffusivity and the magnitude of the convective velocity, the new variational formulation and the corresponding Galerkin scheme are well-posed and a Céa estimate can be derived. We establish the rate of convergence when the flux and the concentration are approximated, respectively, by Raviart–Thomas/Brezzi–Douglas–Marini and continuous piecewise polynomials. In addition, we develop an a posteriori error analysis of residual type. We derive a simple a posteriori error indicator and prove that it is reliable and locally efficient. Finally, we provide some numerical experiments that illustrate the performance of the method.

A semi-Lagrangian discontinuous Galerkin (DG) – local DG method for solving convection-diffusion equations

In this paper, we propose an efficient high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for solving linear convection-diffusion equations. The method generalizes our previous work on developing the SLDG method for transport equations [5], making it capable of handling additional diffusion and source terms. Within the DG framework, the solution is evolved along the characteristics; while the diffusion term is discretized by the local DG (LDG) method and integrated along characteristics by implicit Runge-Kutta methods together with source terms. The proposed method is named the ‘SLDG-LDG’ method and enjoys many attractive features of the DG and SL methods. These include the uniformly high order accuracy (e.g. third order) in space and in time, compact, mass conservative, and stability under large time stepping size. An L2 stability analysis is provided when the method is coupled with the first order backward Euler discretization. Effectiveness of the method are demonstrated by a group of numerical tests in one and two dimensions.

Superconvergence of local discontinuous Galerkin methods with generalized alternating fluxes for 1D linear convection-diffusion equations

This paper investigates superconvergence properties of the local discontinuous Galerkin methods with generalized alternating fluxes for one-dimensional linear convection-diffusion equations. By the technique of constructing some special correction functions, we prove the (2k + 1)-th-order superconvergence for the cell averages, and the numerical traces in the discrete L2 norm. In addition, superconvergence of orders k + 2 and k + 1 is obtained for the error and its derivative at generalized Radau points. All the theoretical findings are confirmed by numerical experiments.

Numerical inverse Laplace transform for convection-diffusion equations

In this paper a novel contour integral method is proposed for linear convection-diffusion equations. The method is based on the inversion of the Laplace transform and makes use of a contour given by an elliptic arc joined symmetrically to two half-lines. The trapezoidal rule is the chosen integration method for the numerical inversion of the Laplace transform, due to its well-known fast convergence properties when applied to analytic functions. Error estimates are provided as well as careful indications about the choice of several involved parameters. The method selects the elliptic arc in the integration contour by an algorithmic strategy based on the computation of pseudospectral level sets of the discretized differential operator. In this sense the method is general and can be applied to any linear convection-diffusion equation without knowing any a priori information about its pseudospectral geometry. Numerical experiments performed on the Black-Scholes ($1D$) and Heston ($2D$) equations show that the method is competitive with other contour integral methods available in the literature.

Sharp decay estimates in local sensitivity analysis for evolution equations with uncertainties: From ODEs to linear kinetic equations

We review the Lyapunov functional method for linear ODEs and give an explicit construction of such functionals that yields sharp decay estimates, including an extension to defective ODE systems. As an application, we consider three evolution equations, namely the linear convection-diffusion equation, the two velocity BGK model and the Fokker–Planck equation.Adding an uncertainty parameter to the equations and analyzing its linear sensitivity leads to defective ODE systems. By applying the Lyapunov functional construction, we prove sharp long time behavior of order (1+tM)e−μt, where M is the defect and μ is the spectral gap of the system. The appearance of the uncertainty parameter in the three applications makes it important to have decay estimates that are uniform in the non-defective limit.

Superconvergence analysis of local discontinuous Galerkin methods for linear convection–diffusion equations in one space dimension

This paper is concerned with the superconvergence study of the local discontinuous Galerkin (LDG) method for one-dimensional time-dependent linear convection–diffusion equations, where the convection flux is taken as the upwind flux, while the diffusion fluxes chosen as the alternating fluxes. Superconvergence properties for both the solution itself and auxiliary variables are established. Precisely, we prove that, the LDG solutions are superconvergent with an order of \(k+2\) towards a particular projection of the exact solution and the auxiliary variable, and thus a \(k+1\)-th order superconvergence for the derivative approximation and a \(k+2\)-th order superconvergence for the function value approximation at a class of Radau points are obtained. Especially, we show that the convergence rate of the derivative approximation for the exact solution can reach \(k+2\) when the convection flux is the same as the diffusion flux, two order higher than the optimal convergence rate. Furthermore, a \(2k+1\)-th order superconvergent for the errors of the numerical fluxes at mesh nodes as well as for the cell averages, is also obtained under some suitable initial discretization. Numerical experiments indicate that most of our theoretical findings are optimal.

A Domain Decomposition Model Reduction Method for Linear Convection-Diffusion Equations with Random Coefficients

We developed a domain decomposition model reduction method for linear steady- state convection-diffusion equations with random coefficients. Of particular interest to this effort are the diffusion equation with random diffusivity and the convection-dominated transport equation with random velocity. We investigated two types of random fields, i.e., colored noises and discrete white noises, both of which can lead to high-dimensional parametric dependence in practice. The motivation is to exploit domain decomposition to reduce the parametric dimension of local problems in subdomains, such that an entire parametric map can be approximated with a small number of expensive partial differential equation (PDE) simulations. The new method combines domain decomposition with model reduction and sparse polynomial approximation, so as to simultaneously handle the high dimensionality and irregular behavior of the PDEs under consideration. The advantages of our method lie in three aspects: (i) online-offline decomposition, i.e., the online cost is independent of the size of the triangle mesh; (ii) sparse approximation of operators involving nonaffine high-dimensional random fields; (iii) an effective strategy to capture irregular behaviors, e.g., sharp transitions of the PDE solutions. Two numerical examples are provided to demonstrate the advantageous performance of our method.

Error estimates of the third order runge-kutta alternating evolution discontinuous galerkin method for convection-diffusion problems

In this paper, we present the stability analysis and error estimates for the alternating evolution discontinuous Galerkin (AEDG) method with third order explicit Runge-Kutta temporal discretization for linear convection-diffusion equations. The scheme is shown stable under a CFL-like stability condition c0τ ≤ ε ≤ c1h2. Here ε is the method parameter, and h is the maximum spatial grid size. We further obtain the optimal L2 error of order O(τ3 + hk+1). Key tools include two approximation finite element spaces to distinguish overlapping polynomials, coupled global projections, and energy estimates of errors. For completeness, the stability analysis and error estimates for second order explicit Runge-Kutta temporal discretization is included in the appendix.

A non-overlapping DDM combined with the characteristic method for optimal control problems governed by convection–diffusion equations

In this paper, we consider a non-overlapping domain decomposition method combined with the characteristic method for solving optimal control problems governed by linear convection–diffusion equations. The whole domain is divided into non-overlapping subdomains, and the global optimal control problem is decomposed into the local problems in these subdomains. The integral mean method is utilized for the diffusion term to present an explicit flux calculation on the inter-domain boundary in order to communicate the local problems on the interfaces between subdomains. The convection term is discretized along the characteristic direction. We establish the fully parallel and discrete schemes for solving these local problems. A priori error estimates in $$L^2$$ -norm are derived for the state, co-state and control variables. Finally, we present numerical experiments to show the validity of the schemes and verify the derived theoretical results.

Stability Analysis and Error Estimates of Semi-implicit Spectral Deferred Correction Coupled with Local Discontinuous Galerkin Method for Linear Convection–Diffusion Equations

In this paper, we focus on the theoretical analysis of the second and third order semi-implicit spectral deferred correction (SDC) time discretization with local discontinuous Galerkin (LDG) spatial discretization for the one-dimensional linear convection–diffusion equations. We mainly study the stability and error estimates of the corresponding fully discrete scheme. Based on the Picard integral equation, the SDC method is driven iteratively by either the explicit Euler method or the implicit Euler method. It is easy to implement for arbitrary order of accuracy. For the semi-implicit SDC scheme, the iteration and the left-most endpoint involved in the integral for the implicit part increase the difficulty of the theoretical analysis. To be more precise, the test functions are more complex and the energy equations are more difficult to construct, compared with the Runge–Kutta type semi-implicit schemes. Applying the energy techniques, we obtain both the second and third order semi-implicit SDC time discretization with LDG spatial discretization are stable provided the time step $$\tau \le \tau _{0}$$ , where the positive $$\tau _{0}$$ depends on the diffusion and convection coefficients and is independent of the mesh size h. We then obtain the optimal error estimates for the corresponding fully discrete scheme under the condition $$\tau \le \tau _{0}$$ with similar technique for stability analysis. Numerical examples are presented to illustrate our theoretical results.