Convection-diffusion Partial Differential Equations Research Articles

Numerical approximation of parabolic singularly perturbed problems with large spatial delay and turning point

PurposeSingular perturbation turning point problems (SP-TPPs) involving parabolic convection–diffusion Partial Differential Equations (PDEs) with large spatial delay are studied in this paper. These type of equations are important in various fields of mathematics and sciences such as computational neuroscience and require specialized techniques for their numerical analysis.Design/methodology/approachWe design a numerical method comprising a hybrid finite difference scheme on a layer-adapted mesh for the spatial discretization and an implicit-Euler scheme on a uniform mesh in the temporal variable. A combination of the central difference scheme and the simple upwind scheme is used as the hybrid scheme.FindingsConsistency, stability and convergence are investigated for the proposed scheme. It is established that the present approach has parameter-uniform convergence of OΔτ+K−2(lnK)2, where Δτ and K denote the step size in the time direction and number of mesh-intervals in the space direction.Originality/valueParabolic SP-TPPs exhibiting twin boundary layers with large spatial delay have not been studied earlier in the literature. The presence of delay portrays an interior layer in the considered problem’s solution in addition to twin boundary layers. Numerical illustrations are provided to demonstrate the theoretical estimates.

High-order entropy stable schemes based on a local adaptive WENO viscosity for degenerate convection-diffusion equations

In this work, high-order entropy stable finite difference methods are constructed to solve systems of degenerate convection-diffusion partial differential equations. The approximation of the convective term is based on high-order entropy stable fluxes under a TeCNO scheme formulation and for the diffusive term high-order entropy conservative fluxes are computed by considering linear combinations of second-order approximations. Numerical simulations show that the proposed high-order entropy conservative schemes provide a convergence rate better than conventional WENO methods. To obtain entropy stability, a local adaptive artificial viscosity is included using weighted essentially nonoscillatory reconstructions satisfying a local sign property at discontinuities. Different WENO interpolations for the local artificial viscosity are numerically analyzed being the fast and optimal WENO interpolation (FOWENO), the one that introduces less spurious oscillations with a lower computational cost.

Bayesian estimation and reconstruction of marine surface contaminant dispersion

Discharge of hazardous substances into the marine environment poses a substantial risk to both public health and the ecosystem. In such incidents, it is imperative to accurately estimate the release strength of the source and reconstruct the spatio-temporal dispersion of the substances based on the collected measurements. In this study, we propose an integrated estimation framework to tackle this challenge, which can be used in conjunction with a sensor network or a mobile sensor for environment monitoring. We employ the fundamental convection-diffusion partial differential equation (PDE) to represent the general dispersion of a physical quantity in a non-uniform flow field. The PDE model is spatially discretised into a linear state-space model using the dynamic transient finite-element method (FEM) so that the characterisation of time-varying dispersion can be cast into the problem of inferring the model states from sensor measurements. We also consider imperfect sensing phenomena, including miss-detection and signal quantisation, which are frequently encountered when using a sensor network. This complicated sensor process introduces nonlinearity into the Bayesian estimation process. A Rao-Blackwellised particle filter (RBPF) is designed to provide an effective solution by exploiting the linear structure of the state-space model, whereas the nonlinearity of the measurement model can be handled by Monte Carlo approximation with particles. The proposed framework is validated using a simulated oil spill incident in the Baltic sea with real ocean flow data. The results show the efficacy of the developed spatio-temporal dispersion model and estimation schemes in the presence of imperfect measurements. Moreover, the parameter selection process is discussed, along with some comparison studies to illustrate the advantages of the proposed algorithm over existing methods.

MIXED VOLUME ELEMENT-CHARACTERISTIC MIXED VOLUME ELEMENT OF NUMERICAL SIMULATION OF NUCLEAR CONTAMINATION TREATMENT AND ITS CONVERGENCE ANALYSIS

We consider a nonlinear system with boundary-initial value conditions of convection-diffusion partial differential equations describing nuclear waste disposal contamination in porous media. The flow pressure is determined by an elliptic equation, the concentrations of brine and radionuclide are formulated by convection-diffusion equations, and the transport of temperature is defined by a heat equation. The transport pressure appears in the concentration equations and heat equation accompanying with Darcy velocity, and controls their processes. The flow equation is solved by the conservative method of mixed volume element and the accuracy of Darcy velocity is improved one order. The method of characteristic mixed volume element is applied to solve the concentrations and the heat, where the diffusion is discretized by a mixed volume element method and the convection is treated by the method of characteristics. The characteristics can confirm high computation stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. The scheme can adopt a large step while it has smaller time-truncation error and higher order of accuracy. The mixed volume element method has law of conservation on every element to treat the diffusion and it can obtain numerical solution is of the concentration and adjoint vectors. Using the theory and technique of priori estimate of differential equations, we derive an optimal second order estimate in l2 norm. Numerical examples are given to show the effectiveness and practicability and the method is testified as a powerful tool to solve such an international famous problem.

A deep learning feed-forward neural network framework for the solutions to singularly perturbed delay differential equations

In this paper, we explore a deep learning feedforward artificial neural network (ANN) framework as a numerical tool for approximating the solutions to singularly perturbed delay differential equations (SPDDE). Our approach trains the network with fewer uniform data points along with linear interpolation as a dependent variable. More importantly, the exact solution is not used for training the network. A mean square or Euclidean norm type total loss function is utilized to assess the deep learning network’s performance. In the training and testing phase, our network adjusts itself to fit the tangent and the curvature by minimizing the total loss function and fine-tuning the network’s hyperparameters during the backpropagation stage. Our focus in this contribution is to investigate an answer to the question- “whether neural networks can learn to solve the differential equations with both delay and boundary layer behavior”? Traditional numerical methods are known to perform poorly in approximating the solutions to SPDDE due to boundary layer behavior, which is the sharp change in the gradient of the solution inside a region of very small width. Our proposed network architecture is used to investigate the above question for three varieties of SPDDE. The numerical results demonstrate that the fine-tuned adaptive deep learning architecture can effectively approximate the solution to SPDDE for varieties of delay and perturbation parameters. It is expected that the current method can be extendable to approximate 2D convection-diffusion partial differential equations with a boundary layer.

A Fourier-based methodology without numerical diffusion for conducting dye simulations and particle residence time calculations

Dye experimentation is a widely used method in experimental fluid mechanics for flow analysis or for the study of the transport of particles within a fluid. This technique is particularly useful in biomedical diagnostic applications ranging from hemodynamic analysis of cardiovascular systems to ocular circulation. However, simulating dyes governed by convection-diffusion partial differential equations (PDEs) can also be a useful post-processing analysis approach for computational fluid dynamics (CFD) applications. Such simulations can be used to identify the relative significance of different spatial subregions in particular time intervals of interest in an unsteady flow field. Additionally, dye evolution is closely related to non-discrete particle residence time (PRT) calculations that are governed by similar PDEs. This contribution introduces a pseudo-spectral method based on Fourier continuation (FC) for conducting dye simulations and non-discrete particle residence time calculations without numerical diffusion errors. Convergence and error analyses are performed with both manufactured and analytical solutions. The methodology is applied to three distinct physical/physiological cases: 1) flow over a two-dimensional (2D) cavity; 2) pulsatile flow in a simplified partially-grafted aortic dissection model; and 3) non-Newtonian blood flow in a Fontan graft. Although velocity data is provided in this work by numerical simulation, the proposed approach can also be applied to velocity data collected through experimental techniques such as from particle image velocimetry.

Learning subgrid-scale models with neural ordinary differential equations

We propose a new approach to learning the subgrid-scale model when simulating partial differential equations (PDEs) solved by the method of lines and their representation in chaotic ordinary differential equations, based on neural ordinary differential equations (NODEs). Solving systems with fine temporal and spatial grid scales is an ongoing computational challenge, and closure models are generally difficult to tune. Machine learning approaches have increased the accuracy and efficiency of computational fluid dynamics solvers. In this approach neural networks are used to learn the coarse- to fine-grid map, which can be viewed as subgrid-scale parameterization. We propose a strategy that uses the NODE and partial knowledge to learn the source dynamics at a continuous level. Our method inherits the advantages of NODEs and can be used to parameterize subgrid scales, approximate coupling operators, and improve the efficiency of low-order solvers. Numerical results with the two-scale Lorenz 96 ODE, the convection–diffusion PDE, and the viscous Burgers’ PDE are used to illustrate this approach.

A Model of Froth Flotation with Drainage: Simulations and Comparison with Experiments

The operation of a froth flotation column can be described by a nonlinear convection–diffusion partial differential equation that incorporates the solids–flux and drift–flux theories as well as a model of foam drainage. The resulting model predicts the bubble and (gangue) particle volume fractions as functions of height and time. The steady-state (time-independent) version of the model defines so-called operating charts that map conditions on the gas and pulp feed rates that allow for operation with a stationary froth layer. Operating charts for a suitably adapted version of the model are compared with experimental results obtained with a laboratory flotation column. Experiments were conducted with a two-phase liquid–bubble flow. The results indicate good agreement between the predicted and measured conditions for steady states. Numerical simulations for transient operation, in part for the addition of solid particles, are presented.

New Fifth-Kind Chebyshev Collocation Scheme for First-Order Hyperbolic and Second-Order Convection-Diffusion Partial Differential Equations

The fifth type of Chebyshev polynomials was used in tandem with the spectral tau method to achieve a semianalytical solution for the partial differential equation of the hyperbolic first order. For this purpose, the problem was diminished to the solution of a set of algebraic equations in unspecified expansion coefficients. The convergence and error analysis of the proposed expansion were studied in-depth. Numerical trials have confirmed the applicability and the accuracy.

Numerical Study on an RBF-FD Tangent Plane Based Method for Convection-Diffusion Equations on Anisotropic Evolving Surfaces.

In this paper, we present a fully Lagrangian method based on the radial basis function (RBF) finite difference (FD) method for solving convection–diffusion partial differential equations (PDEs) on evolving surfaces. Surface differential operators are discretized by the tangent plane approach using Gaussian RBFs augmented with two-dimensional (2D) polynomials. The main advantage of our method is the simplicity of calculating differentiation weights. Additionally, we couple the method with anisotropic RBFs (ARBFs) to obtain more accurate numerical solutions for the anisotropic growth of surfaces. In the ARBF interpolation, the Euclidean distance is replaced with a suitable metric that matches the anisotropic surface geometry. Therefore, it will lead to a good result on the aspects of stability and accuracy of the RBF-FD method for this type of problem. The performance of this method is shown for various convection–diffusion equations on evolving surfaces, which include the anisotropic growth of surfaces and growth coupled with the solutions of PDEs.

Simulations of liquid chromatography using two-dimensional non-equilibrium lumped kinetic model with bi-Langmuir isotherm

A two-dimensional non-equilibrium and non-linear lumped kinetic model of liquid chromatography is formulated and numerically approximated to simulate the separation of multi-component mixtures in a packed fixed bed cylindrical column operating under isothermal conditions. The model equations incorporate the bi-Langmuir adsorption thermo-dynamics as well as the radial and axial variations of concentration. By introducing distinct regions of injection at the column inlet, radial concentration gradients are generated to intensify the effect of mass transfer rate in the radial-direction, inside the column. The mathematical model is developed by a system of non-linear convection-diffusion partial differential equations for mass balance in the mobile phase, coupled with differential equation for mass balance in the stationary phase and algebraic equations for adsorption isotherm. In this study, a high-resolution, semi-discrete, finite-volume technique is formulated and applied to gain the numerical solution of the governing non-linear-model equations. A few numerical case studies are performed to investigate the effects of the various critical parameters on the process performance. The developed numerical algorithm provide an efficacious mechanism for investigating the retention behavior, systematic monitoring and efficient operation of non-equilibrium, liquid chromatographic processes.

A semi-Lagrangian meshfree Galerkin method for convection-dominated partial differential equations

A new numerical method, which is based on the coupling between semi-Lagrangian (SL) method and element free Galerkin (EFG) method, is developed for convection–diffusion partial differential equations with dominated convection terms. It is the first time that the SL method combined with the EFG method to solve the transient convection–diffusion equations. This combination has been more effective than expected. Compared with the existing SL methods, the proposed method shows several new advantages. Firstly, it takes full advantages of meshfree methods, therefore, the implementation difficulties of the SL method in backward tracing and departure point interpolation are greatly reduced. As interpolating the function values to a departure point, complex searching algorithm is not needed to determine which grid cell contains the departure point, and higher order interpolation is not essential to ensure the global accuracy. Meanwhile, in addition to having good stability and allowing the use of large time-step sizes, the new method realizes the continuous transition from convection–diffusion equation to pure convection equation and provides a uniform numerical format for these two kinds of equations. A problem with available analytical solution is solved to analyze the convergence behavior of the proposed method. It is demonstrated that the method can achieve the optimal convergence rate. Subsequently, the method is applied to solve the 1D and 2D Burgers’ equations. The results show that the proposed method can indeed obtain surprisingly good results for the Burgers’ equations in the very challenging convection-dominated and pure convection cases. These results are better than almost all the ones we have seen in literature. The good performance of this method in solving the challenging Burgers’ equations makes us believe that it will be a very promising method for solving convection-dominated problems.

A Hybrid Differential Transforms and Finite Difference Method to Numerical Solution of Convection–Diffusion Equation

In this work, we discuss a hybrid-based method on differential transforms and a finite difference method to numerical solution of convection–diffusion equation with Dirichlet’s type boundary conditions. The developed method is tested on various problems and the numerical results are reported in tabular and figure form. This method can be easily extended to handle non-linear convection–diffusion partial differential equations.

Numerical Simulation of the Convection–Diffusion PDEs on a Sphere with RBF-FD and RBF-QR Methods

The aim of this paper is to investigate the application of radial basis function-generated finite difference (RBF-FD) methods for convection–diffusion partial differential equations (PDEs) on a sphere. In the application of RBF-FD method, choosing a reasonable value of shape parameter is important to the computation of PDEs. The work is devoted to the numerical study of the range of near optimal shape parameters for the convection–diffusion equations. Because the RBF-FD Direct method often leads to ill-conditioned problems for small shape parameters, the RBF-QR method is applied locally to overcome the ill-conditioning in the context of RBF-FD mode. Additionally, for convection-dominated problems, it can be found that the results of using central-type stencil present spurious oscillations. Therefore, we propose an upwind RBF-FD (URBF-FD) scheme to overcome the problems, which is well adapted to the problems on the sphere and easy to be implemented. Further numerical results show that the proposed URBF-FD method is stable and effective for convection-dominated PDEs on the sphere.

Discontinuous-Galerkin finite-element method for approximating a model of non-equilibrium liquid chromatography considering Bi-Langmuir isotherm

In this paper, a discontinuous-Galerkin finite-element method (DG-FEM) is applied for the numerical solution of multi-component, non-equilibrium, lump kinetic model of liquid chromatography under nonlinear conditions. The model is analyzed for standard Bi-Langmuir type adsorption isotherms using Danckwert boundary conditions. Packed bed processes of liquid chromatography are modeled as convection-diffusion partial differential equations (PDE). In these models, the diffusion term is strongly dominated by the convection term. Therefore simulation of packed bed chromatographic processes requires specialized numerical techniques. In this study, the (DG-FEM) is utilized for the space discretization and the resulting semi-discrete system of ordinary differential equations is solved numerically by using a total variation bounded (TVB) Runge–Kutta method. This technique resolves sharp discontinuities and achieves a high order accuracy. To inspect the impact of different parameters, the results of the proposed method are authenticated against the high-resolution finite-volume scheme (HR-FVS). These numerical results include a single-solute flow, two-component mixture flow, and three-component mixture flow. The developed numerical results could be helpful in optimal predictive control, systematic monitoring and efficient operation of chromatographic processes.

Discontinuous Galerkin scheme for solving non-isothermal and non-equilibrium model of liquid chromatography

A total variation bounded (TVB) Runge–Kutta local-projection discontinuous Galerkin (RK-LDG) finite element method is applied for numerically approximating the non-isothermal and non-equilibrium model of liquid chromatography. The model comprises a system of convection-diffusion partial differential equations (PDEs) coupled with algebraic and ordinary differential equations (ODEs). The suggested numerical scheme is explicit in nature and has the potential to capture sharp peaks and abrupt changes in the solution profiles. A number of case studies are carried out for single and two-component elution. The influence of temperature on the column efficiency and separation of mixture components is analyzed. Furthermore, important parameters are identified that influence the performance of the column. The results of the proposed method are authenticated against the high-resolution finite-volume scheme (HR-FVS).

Numerical simulation of various kind of convection-diffusion partial differential equation

Numerical simulation of various kind of convection-diffusion partial differential equation

An interpolation method for the optimal control problem governed by the elliptic convection–diffusion equation

AbstractWe know that due to the Weierstrass approximation theorem any continuous function over a closed interval can be approximated by a polynomial of sufficiently high degree. Therefore, they are frequently used to approximate functions. We proposed a Lagrange polynomial‐based approach for solving the optimal control problem governed by the elliptic convection–diffusion partial differential equation. This paper solves it by the barycentric interpolation method as a class of strong‐form numerical methods. Thanks to Lagrangian multipliers, optimality system is derived and then the barycentric collocation method is employed to discretize the state variable and adjoint variable. Barycentric interpolation method is a Lagrange polynomial interpolation that is fast and deserves to be known as a method of polynomial interpolation. The convergence of the proposed method is also proved. At the end, numerical experiments are employed to illustrate the theoretical findings.

Contrast enhanced tomographic reconstruction of vascular blood flow based on the Navier-Stokes equation.

ABSTRACT In this work, we study the reconstruction of blood velocity with contrast enhanced computed tomography with tomographic projections perpendicular to the main flow field direction. The inverse problem is regularized with a convection-diffusion partial differential equation and with the Navier-Stokes equation. The velocity field is reconstructed together with the density field with a first-order adjoint method. The method is validated on a simple phantom. The addition of a physical constraint with a partial differential equation for the velocity field improves the reconstruction results.

A non-uniform difference scheme for solving singularly perturbed 1D-parabolic reaction–convection–diffusion systems with two small parameters and discontinuous source terms

This paper aims at solving numerically the 1-D weakly coupled system of singularly perturbed reaction–convection–diffusion partial differential equations with two small parameters and discontinuous source terms. Boundary and interior layers appear in the solutions of the problem for sufficiently small values of the perturbation parameters. A numerical algorithm based on finite difference operators and an appropriate piecewise uniform mesh is constructed and its characteristics are analyzed. The method is confirmed to reach almost first order convergence, independently of the values of the perturbation parameters. Some numerical experiments are presented, which serve to illustrate the theoretical results.