Special Session 135: Latest Developments in Computational Methods for Differential Equations Arising in Fluid Dynamics with Multi-scale and Boundary Layer Behaviour

Singularly Perturbed Differential Equations on a Graph Domain

Vivek Kumar Aggarwal Delhi Technological University India Co-Author(s):

Abstract: In this talk, singularly perturbed convection dominant and reaction diffusion problems will be discussed on a graph domain. Challenges (such as Kirchhoff`s conditions, continuity conditions) related to the graph domain will be taken care. Some problems on a tripod is solved numerically and in closed form for the validation of the proposed method. A general graph is also discussed along with a diamond graph for singularly perturbed problems. Error estimated have also been done.

Data Driven Approach to Estimate Perturbation Parameter for Singularly Perturbed Problems using Differential Evolution

Vivek Kumar Aggarwal Delhi Technological University India Co-Author(s):

Abstract: This paper introduces a machine learning framework using differential evolution algorithm to learn the perturbation parameter ($\epsilon^{\alpha}$) involved in the singularly perturbed problems. Perturbation parameter plays a vital role in the field of the perturbation theory and is responsible for generating boundary/interior layers. Initially, random data is being used as an population for the estimation of the parameter ($\alpha$). Two types of problems: convection diffusion and reaction diffusion, have been tested for the estimation of the parameter. For comparison, the same test problems have also been solved using the particle swarm optimization method. For the values of the various parameters, results have been shown in the tables along with best cost figures.

Stiff Order Conditions in Runge-Kutta Methods for Linear and Semi-Linear Problems

Abhijit Biswas king Abdullah University of Science and Technology Saudi Arabia Co-Author(s):    David Ketcheson, Steven Roberts, Benjamin Seibold, David Shirokoff

Abstract: Runge-Kutta (RK) methods may demonstrate order reduction when applied to stiff problems. This talk explores the issue of order reduction in Runge-Kutta methods specifically when dealing with linear and semi-linear stiff problems. First, I will introduce Diagonally Implicit Runge-Kutta (DIRK) methods with high Weak Stage Order (WSO), capable of mitigating order reduction in linear problems with time-independent operators. On the theoretical front, I will present order barriers relating the WSO of an RK scheme to its order and the number of stages for fully-implicit RK and DIRKmethods, serving as a foundation to construct schemes with high WSO. I will conclude by presenting stiff order conditions for semilinear problems, essential to extend beyond the limitations of WSO, which primarily focused on linear problems.

A parameter uniform hybrid approach for singularly perturbed two-parameter parabolic problem with discontinuous data

Anuradha Jha Indian Institute of Information Technology Guwahati India Co-Author(s):    Nirmali Roy, Anuradha Jha

Abstract: In this talk, we discuss singularly perturbed two-parameter 1D parabolic problem of the reaction-convection-diffusion type. These problems exhibit discontinuities in the source term and convection coefficient at particular domain points, which results in the interior layers. Presence of perturbation parameters give rise to the boundary layers too. To resolve these layers a hybrid monotone difference scheme is used on a piece-wise uniform Shishkin mesh in the spatial direction and Crank-Nicolson scheme is used on a uniform mesh in the temporal direction. The resulting scheme is proven to be almost second order convergent in spatial direction and order two in temporal direction. Numerical experiments corroborate the theoretical claims made.

Layer-Resolving Numerical Methods for Degenerate Singular Perturbations Problems with Two Parameters

Anirban Majumdar Indian Institute of Information Technology Design and Manufacturing Kurnool India Co-Author(s):    Mrityunjoy Barman, Natesan Srinivasan, Anirban Majumdar

Abstract: This work addresses a class of steady-state and time-dependent degenerate singular perturbation problems with two parameters affecting the convection and diffusion terms. Due to the presence of degeneracy and multiple perturbation parameters, the continuous solution exhibits boundary layers with different widths at the boundaries of the spatial domain. To effectively capture these layers, we utilize a piecewise uniform Shishkin grid for spatial discretization and a uniform grid for time discretization. The time derivative is approximated using an implicit Euler method on the equispaced temporal grid, while upwind finite difference schemes are applied to the Shishkin mesh for spatial derivatives. To enhance solution accuracy, we incorporate the Richardson extrapolation technique. Our theoretical analysis establishes an error bound, demonstrating almost second-order convergence. Numerical experiments are conducted to corroborate the theoretical findings, confirming the predicted convergence rates.

A Newton linearized two-level alternating direction implicit scheme for two-dimensional time fractional reaction--diffusion equation exhibiting weak initial singularity

Deeksha Singh SRMIST KTR, Chennai India Co-Author(s):    Deeksha Singh, Rajesh K. Pandey

Abstract: A two-dimensional nonlinear time fractional reaction--diffusion equation of order $\alpha\in (0, 1)$ is considered. The typical solution to such problems usually has an initial layer at $t=0.$ To capture the initial singularity, the Caputo time fractional derivative is approximated using the $L2-1_{\sigma}$ formula on the smoothly graded meshes. Spatial derivatives are approximated using standard central difference approximation. Computational cost is reduced using Newton`s linearized method and alternating direction implicit method. Theoretical analysis comprising Solvability, stability, and convergence of the finite difference scheme has been studied rigorously and it is shown that the method is convergent with convergence order $\mathcal{O}(M^{-\min\{3-\alpha, r\alpha, 1+\alpha, 2+\alpha\}}+h_{x}^{2}+h_{y}^{2})$ where $M$ is the temporal discretization parameter, $h_{x}$, $h_{y}$ are the step sizes in the spatial direction and $\alpha\in (0,1)$ is the fractional order. The applicability of the discussed numerical scheme is established by two illustrative examples having smooth and nonsmooth solutions.

Direct discontinuous Galerkin method for two parameter singular perturbation problems

Gautam Singh National Institute of Technology Tiruchirappalli India Co-Author(s):

Abstract: Many disciplines of practical mathematics, such as elasticity, fluid dynamics, reaction-diffusion processes, geophysics, chemical reactor theory, and so on, have singular perturbation problems (SPPs). It is interesting to note that the SPPs solution demonstrates a multi-scale character, that is, there are thin layers where the solution varies quickly, while away from the layer(s) the solution varies slowly. So, due to the thin layer(s) in the solution of SPPs, there are many computational difficulties in the numerical solution of SPPs. To address this class of problems, numerous variations of discontinuous Galerkin (DG) methods have been used. The DG technique is a popular finite element technique that uses a completely discontinuous polynomial space for numerical solution and test functions. A key ingredient of the DG methods is the suitable design of numerical fluxes to obtain high-order accuracy and stable schemes. Many stunning qualities have contributed to the rise in popularity of these technologies in recent years. Allowing for arbitrary domain decompositions, high-order numerical precision, total freedom in modifying the polynomial degrees in each element independent of the neighbours, and an extremely local data structure resulting in excellent parallel efficiency are some of the features of the DG methods. In this talk, We propose a direct discontinuous Galerkin (DDG) methods for two-paramter singular perutrbation problems. We have shown that method is uniformly convergent with the order $k$ in energy norm, where $k$ is the degree of piecewise polynomial in finite element space. We have presented numerical result to verify our theoretical findings.

An Efficient Robust Computational Method for Singularly Perturbed 1D Parabolic PDEs

Natesan Srinivasan Indian Institute of Technology Guwahati India Co-Author(s):    Suraj Kumar

Abstract: Here, we study the numerical solution of singularly perturbed 1D parabolic PDE exhibiting boundary layers. Crank-Nicolson scheme is used to discretize the time derivative on uniform mesh and Non-symmetric Interior Penalty Galerkin (NIPG) is used for the spatial derivatives on the exponentially-graded mesh. Stability, convergence and superconvergence are studied and numerical experiments are provided.

Numerical Solution of Two-Parameter Singularly Perturbed Differential Equations by Efficient Physics-Informed Neural Networks

Natesan Srinivasan Indian Institute of Technology Guwahati India Co-Author(s):    Aayushman Raina and Pradanya Boro

Abstract: In recent years, machine learning techniques, namely, physics informed neural networks (PINNs) are becoming popular to solve various types of differential equations modelling several physical phenomena. Here, we focus the fundamentals of PINNs, and, their performance to solve 1D and 2D singularly perturbed differential equations having two small parameters in the diffusion and convection terms. Further, we study the shortfalls of classical PINNs in solving two-parameter singular perturbation problems, and how to overcome these difficulties through other variants of PINNs. Several numerical experiments are carried out to see their performance.

Robust conservative finite element methods for incompressible flows: with lower degrees

Shuo Zhang Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):

Abstract: The strict preservation of the conservation property is important in the design of numerical schemes for various model problems. I will firstly talk about why we would like to study low-degree strictly conservative finite element method for incompressible flows. Then I will talk about a nonstandard approach for designing finite element schemes for fluid computation, which can preserve strictly the divergence free condition for incompressible fluid flows. The schemes work on general triangulations with lower degree of polynomials than known results, and its superiority with respect to some existing schemes are partially illustrated with numerical experiments, including ones with boundary layers. The theoretical analysis depends on a careful application of Stokes complex. Both boundary value problems and eigenvalue problems will be mentioned, in case the time permits.