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[CT102]

[00032] A geometrically preservative semi-adaptive method for the numerical solution of Kawarada equations

  • Session Date & Time : 1E (Aug.21, 17:40-19:20)
  • Type : Contributed Talk
  • Abstract : This presentation concerns the numerical stability and geometric preservations of the numerical solution of Kawarada equation problems. The nonlinear partial differential equations exhibit strong quenching types of singularities that represent a number of key characteristics from industrial and multi-physical applications. A second order semi-adaptive implicit finite difference method will be constructed and investigated. We shall begin with a detailed mathematical analysis of the stability without freezing singular source terms of Kawarada equations in this talk. Preservation features of the solution vector sequences will then be studied. Realistic orders of the convergence will be given via generalized Milne's devices. Finally, computer simulations will be carried out to demonstrate the effectiveness of the theoretical analysis and conclusions.
  • Classification : 65M06, 65M12, 65M50, 68U01, 65D18
  • Author(s) :
    • Qin Sheng (Baylor University)

[00104] High order approximation of Caputo-Prabhakar derivative and its application in solving time fractional Advection-Diffusion equation

  • Session Date & Time : 1E (Aug.21, 17:40-19:20)
  • Type : Contributed Talk
  • Abstract : This work aims to devise a high-order numerical scheme to approximate the CaputoPrabhakar derivative of order 0 < α < 1, using an rth degree Lagrange interpolation polynomial, where $3\leq r\in\mathbb{Z^{+}}.$. This numerical scheme can be thought of as an extension of the presented schemes for the approximation of the Caputo-Prabhakar derivative in our previous work \cite{r1}. Further, we adopt the proposed scheme to solve a time-fractional Advection-Diffusion equation with the Dirichlet boundary condition. It is shown that the method is unconditionally stable, uniquely solvable, and convergent with convergence order, $ O(\tau^{r+1-\alpha}, h^{2}), $ where τ and h are the step sizes in the temporal and spatial directions, respectively. Without loss of generality, obtained results are supported by numerical examples for r = 4, 5. \bibitem{r1} Deeksha Singh, Farheen Sultana, and Rajesh K Pandey, Approximation of Caputo Prabhakar derivative with application in solving time fractional advection-diffusion equation, International Journal for Numerical Methods in Fluids. $94(7)(2022)$, pp. 896-919.
  • Classification : 65M06, 65M12, Numerical approximation of fractional derivative and its application
  • Author(s) :
    • DEEKSHA SINGH (Department of Mathematical Sciences, Indian Institute of Technology, BHU, Varanasi)
    • Rajesh K. Pandey (Department of Mathematical Sciences, Indian Institute of Technology, BHU, Varanasi)

[00111] A convergent numerical method for time-fractional reaction-diffusion equation

  • Session Date & Time : 1E (Aug.21, 17:40-19:20)
  • Type : Contributed Talk
  • Abstract : This paper design and analyze a robust finite difference scheme for solving a time-fractional reaction-diffusion equation with smooth and non-smooth solutions. The solution of this equation exhibits a weak singularity at the initial time $\mathrm{t}=0$. So we use graded temporal mesh in order to handle the singularity. We discretize the space variable using a cubic polynomial spline difference scheme. Further, the stability and convergence for both the smooth and non-smooth solutions are analyzed separately.
  • Classification : 65M06, 65M12
  • Author(s) :
    • Anshima Singh (Indian Institute of Technology (BHU) Varanasi)

[00314] The Orthogonal Spline Collocation Method for Parabolic Problems with Interfaces

  • Session Date & Time : 1E (Aug.21, 17:40-19:20)
  • Type : Contributed Talk
  • Abstract : The parabolic problems with interfaces are solved using a method in which orthogonal spline collocation (OSC) is employed for the spatial discretization and the Crank–Nicolson method for the time-stepping. The derivation of the method is described in detail for the case in which cubic monomial basis functions are used in the development of the OSC discretization. The results of extensive numerical experiments involving examples from the literature are presented.
  • Classification : 65M06, 65M22, 65M55, 65M70, 35K05
  • Author(s) :
    • Danumjaya Palla (BITS-Pilani KK Birla Goa Campus)
    • Santosh Kumar Bhal (Centurion University of Technology and Management)
    • Graeme Fairweather (Mathematical Reviews, American Mathematical Society)

[00328] Schrödinger map and Multifractality

  • Session Date & Time : 1E (Aug.21, 17:40-19:20)
  • Type : Contributed Talk
  • Abstract : In this talk, we will explore the richness of the Schrödinger map equation by discussing some recent results on its evolution in both hyperbolic and Euclidean geometrical settings. In the latter case, the equivalent form of the equation describes the motion of a vortex filament, e.g., smoke rings, tornadoes, etc. With numerical, theoretical techniques, we will show that when the filament curve initially has corners, its evolution and the trajectory of its corners exhibit multifractality.
  • Classification : 65M06, 28A80, 11L05, 65M20, 35Q55, Mathematical physics, Numerical methods, Schrödinger-type equtaions, Hyperbolic space
  • Author(s) :
    • Sandeep Kumar (CUNEF University)
    • Luis Vega (Basque Center for Applied Mathematics)
    • Francisco de la Hoz (The University of the Basque Country)

[00387] A-WENO Finite-Difference Schemes with a Jordan Based Splitting Flux

  • Session Date & Time : 2C (Aug.22, 13:20-15:00)
  • Type : Contributed Talk
  • Abstract : We propose a third-order A-WENO finite difference scheme for the compressible Euler equations of the gas dynamics. This upwind scheme is based on the flux difference splitting framework using Jordan Canonical Forms. Third-order characteristic-wise WENO-Z interpolations are used to obtain the third-order scheme. The obtained results demonstrate that the new scheme outperforms the Local Lax-Friedrichs scheme in enhancing the resolution of contact waves and fine-scale structures without compromising the robustness.
  • Classification : 65M06, 76M20, 65M08, 76M12, 76N15, Numerical Solutions for Time-Dependent PDEs, Compressible Euler Equations For Gas-Dynamics, Upwind Schemes, Higher-Order Schemes, A-WENO Schemes
  • Author(s) :
    • Naveen Kumar Garg (Indian Institute of Technology Kharagpur, India)
    • Bao-Shan Wang (Ocean University of China)

[00513] The perfectly matched layer for elastic waves in layered media

  • Session Date & Time : 2C (Aug.22, 13:20-15:00)
  • Type : Contributed Talk
  • Abstract : The perfectly matched layer (PML) is widely used to truncate domains in large-scale simulation of wave propagation in open boundaries. PML absorbs outgoing waves without reflection and significantly improves computational efficiency. However, it is very challenging to prove stability of PML models. In this talk, I present our recent contribution on the stability analysis of PML models for the elastic wave equation in layered media modeling seismic wave propagation in the Earth layers.
  • Classification : 65M06, 65M12, 86-08
  • Author(s) :
    • Siyang Wang (Umeå University)

[00906] Efficient Finite-Difference WENO Scheme for Hyperbolic Systems with Non-Conservative Products

  • Session Date & Time : 2C (Aug.22, 13:20-15:00)
  • Type : Contributed Talk
  • Abstract : In this work, we design an adaptive-order finite difference Weighted Essentially Non-Oscillatory (AO FD-WENO) scheme that could accommodate hyperbolic PDEs with non-conservative products. The designing is carried out by writing the scheme in fluctuation form. Because of the use of pointwise primal variables for the update, treatment of the stiff source terms is also very easy for FD-WENO schemes. The designed scheme can capture isolated stationary discontinuities and performs well on test problems requiring well-balancing. For conservation laws, the new FD-WENO is shown to perform as well as the classical version of FD-WENO, with two major advantages:- 1) It can capture jumps in stationary linearly degenerate wave families exactly, 2) It only requires the WENO reconstruction to be applied once. To highlight the versatility of the new method, we have focused on three major hyperbolic systems with non-conservative products: a) The Baer-Nunziato model for compressible multiphase flow, b) The multiphase debris flow model of Pitman and Le and c) The two-layer shallow water equations.
  • Classification : 65M06, 65N06, 35Q35, 76T10, 76M20
  • Author(s) :
    • Dinshaw S Balsara (University of Notre Dame)
    • Deepak Bhoriya (University of Notre Dame)
    • Chi Wang Shu (Brown University)
    • Harish Kumar (Indian Institute of Technology Delhi (IIT-Delhi))

[00999] High-order energy stable schemes for the phase-field model by the Convex Splitting Runge-Kutta methods

  • Session Date & Time : 2C (Aug.22, 13:20-15:00)
  • Type : Contributed Talk
  • Abstract : The Convex Splitting Runge-Kutta method is a high-order energy stable scheme for gradient flow which is a combination of the well-known convex splitting method and the multi-stage Runge-Kutta method. In this talk, we will discuss the applications and challenges of CSRK via extensive examples of the phase-field model.
  • Classification : 65M06, 65M12, 65M70, Phase-field model, Convex splitting method, Runge-Kutta method
  • Author(s) :
    • Jaemin Shin (Chungbuk National University)
    • Hyun Geun Lee (Kwangwoon University)
    • June-Yub Lee (Ewha Womans University)

[01162] Higher-Order Numerical Approximation of Coupled System of Singularly Perturbed Time-Delay Parabolic PDEs on Generalized Adaptive Mesh

  • Session Date & Time : 2C (Aug.22, 13:20-15:00)
  • Type : Contributed Talk
  • Abstract : We consider a coupled system of singularly perturbed convection-diffusion parabolic PDEs with time-delay and having small diffusion parameters of different orders of magnitude. Due to the occurrence of the overlapping boundary layers, generating efficient numerical approximation becomes a challenging task. To accomplish this goal, the governing system of equations is approximated on a generalized S-mesh, a general form of the piecewise-uniform Shishkin mesh, by employing an implicit-Euler method in the time direction together with the upwind finite difference scheme in the spatial direction. At first, we prove that the numerical solution converges uniformly in maximum-norm with faster convergence rate on the generalized S-mesh than the standard Shishkin mesh. Afterwards, we establish higher-order uniform convergence of the resulting numerical solution by applying the Richardson extrapolation technique. The theoretical outcomes are finally supported by the extensive numerical experiments.
  • Classification : 65M06, 65M12, 65M15
  • Author(s) :
    • KAUSHIK MUKHERJEE (Indian Institute of Space Science and Technology (IIST), Thiruvananthapuram)
    • Sonu Bose (Indian Institute of Space Science and Technology (IIST), Thiruvananthapuram)

[01248] Reduction of Computational Cost with Optimal Accurate Approximation for Boundary Layer Originated Two Dimensional Coupled System of Convection Diffusion Reaction Problems

  • Session Date & Time : 2D (Aug.22, 15:30-17:10)
  • Type : Contributed Talk
  • Abstract : In this talk, I will consider a generalized form of a coupled system of time dependent convection diffusion reaction problems having arbitrary small diffusion terms, which lead to boundary layers. The numerical approximations of these problems require adaptive mesh generation for uniformly convergent approximation. In the present talk, I will provide an algorithm which will reduce the computational cost of the system solver by converting the system of discrete equations to a tridiagonal matrix form. This approach together with an adaptive mesh generation technique will preserve the optimal convergence accuracy. This convergence is proved to be independent of diffusion terms magnitude.
  • Classification : 65M06, 65M50, 65N50, Computational Cost Reduction, Error Analysis, Adaptive Mesh Generation, Coupled System of Time Dependent PDEs, Two Dimension
  • Author(s) :
    • Pratibhamoy Das (Indian Institute of Technology Patna)
    • Pratibhamoy Das (Indian Institute of Technology Patna)
    • Shridhar Kumar (Indian Institute of Technology Patna)

[01634] Mimetic schemes applied to the convection diffusion equation: A numerical comparison.

  • Session Date & Time : 2D (Aug.22, 15:30-17:10)
  • Type : Contributed Talk
  • Abstract : Mimetic Finite Difference Schemes (DFM) are increasingly present in the numerical resolution of transient problems [1] since they are more precise than traditional Finite Difference (DF) schemes. However, there are methods in DF that use appropriate combinations of schemes in different nodes in order to eliminate the numerical spread of the method, [2]. In these cases, DF methods are more accurate than DFM. In this work, we start from the equation of convection-diffusion of an incompressible fluid ∂u/∂t+v·∇u=∇·(K∇u), (1) where u(x, t) represents the unknown of the problem, v(x, t) is the velocity, K is the diffusion tensor, and DFM is defined that eliminates the numerical diffusion presented by traditional DFM schemes. To measure the effectiveness of the mimetic schemes, for different configurations of (1), they are compared with the equivalent schemes in DF with the same order of precision as the DFM; for this purpose, the second-order schemes proposed by [2, 3] are taken. Finally, different comparisons are made to verify the results obtained by the given schemes. References: [1] Castillo J. and Grone R.D. A matrix analysis approach to higher-order approximations for divergence and gradients satisfying a global conservation law. SIAM J. Matrix Anal. Appl., 25(1):128– 142, 2003. [2] Mehdi Dehghan. Weighted finite difference techniques for the one-dimensional advection-diffusion equation. Applied Mathematics and Computation, 147(2):307–319, 2004. [3] Mehdi Dehghan. Quasi-implicit and two-level explicit finite-difference procedures for solving the one-dimensional advection equation. Applied Mathematics and Computation, 167(1):46–67, 2005.
  • Classification : 65M06, 65M99
  • Author(s) :
    • Jorge Ospino (Universidad del Norte)
    • Giovanni Calderon (Universidad Industrial de Santander)
    • Jorge Villamizar (Universidad Industrial de Santander)

[01742] A signed distance function preserving scheme for mean curvature flow and related applications

  • Session Date & Time : 2D (Aug.22, 15:30-17:10)
  • Type : Contributed Talk
  • Abstract : Mean curvature flow is an important research topic in geometry, applied mathematics, and the natural sciences. In this talk, we propose a scheme for solving mean curvature flow and some related problems efficiently and accurately on Cartesian grids. Our method uses the sign distance function defined in a narrowband near the moving interface to represent the evolution of the curve. We derive the equivalent evolution equations of distance function in the narrowband. The novelty of the work is to determine the equivalent evolution equation on Cartesian girds without extra conditions or constraints. The proposed method extends the differential operators appropriately so that the solutions on the narrowband are the distance function of the solution to the original mean flow solution. Furthermore, the extended solution carries the correct geometric information, such as distance and curvature, on Cartesian grids. Consequently, it is possible to adapt the existing numerical methods, for instance, finite difference or WENO scheme, that are developed on the Cartesian grids to solve PDEs on curves. The computational domain is a thin narrowband whose widths are a small constant multiple of uniform Cartesian grid spacing. Some experiments confirm that the proposed method is convergent numerically.
  • Classification : 65M06
  • Author(s) :
    • Chia-Chieh Jay Chu (National Tsing Hua UniversityB)

[01848] A Higher Order Schwarz Domain Decomposition Method for Singularly Perturbed Differential Equation

  • Session Date & Time : 2D (Aug.22, 15:30-17:10)
  • Type : Contributed Talk
  • Abstract : We consider a fourth order singularly perturbed differential equation. To solve the problem, the differential equation is transformed into a coupled system of singularly perturbed differential equations. The original domain is divided into three overlapping subdomains. On the regular subdomain, a hybrid scheme is used, while a compact fourth order difference scheme is used on the two boundary layer subdomains on a uniform mesh. We demonstrate that proposed scheme is nearly fourth order uniformly convergent.
  • Classification : 65M06
  • Author(s) :
    • AAKANSHA AAKANSHA (Indian Institutes of Technology (Banaras Hindu University) Varanasi)

[02229] Efficient numerical methods for time-fractional Black-Scholes equation arising in finance

  • Session Date & Time : 2D (Aug.22, 15:30-17:10)
  • Type : Contributed Talk
  • Abstract : Two numerical schemes to solve time-fractional Black-Scholes PDE governing European options. are proposed. First, fractional derivative is discretized by L1-scheme and spatial derivatives by cubic spline method on uniform mesh. Secondly, we discretize temporal variable by L1-scheme on non-uniform mesh and spatial derivatives by NIPG method on uniform mesh. Stability, convergence and numerical results are carried out. Three European options are priced as application and impact of time-fractional derivative order on option price is shown.
  • Classification : 65M06, 65M12, 65M15
  • Author(s) :
    • Jaspreet Kaur Anand (Indian Institute of Technology Guwahati, Guwahati, Assam)
    • Natesan Srinivasan (Indian Institute of Technology Guwahati, Guwahati, Assam)

[02452] Fifth-order WENO Schemes with Z-type Nonlinear Weights for Hyperbolic Conservation Laws

  • Session Date & Time : 2E (Aug.22, 17:40-19:20)
  • Type : Contributed Talk
  • Abstract : In this talk, we propose the variant Z-type nonlinear weights in the fifth-order weighted essentially non-oscillatory (WENO) finite difference scheme for hyperbolic conservation laws. We take new smoothness indicators and follow the form of Z-type nonlinear weights introduced by Borges et al., leading to fifth order accuracy in smooth regions and sharper approximations around discontinuities. Finally, numerical examples are presented to demonstrate the advantages of the proposed WENO schemes in shock-capturing.
  • Classification : 65M06
  • Author(s) :
    • Jiaxi Gu (Pohang University of Science and Technology)
    • Xinjuan Chen (Jimei University)
    • Jae-Hun Jung (Pohang University of Science and Technology)

[02495] Hybrid WENO schemes for hyperbolic conservation laws with non-convex flux

  • Session Date & Time : 2E (Aug.22, 17:40-19:20)
  • Type : Contributed Talk
  • Abstract : The WENO reconstruction is a high-order accurate method for hyperbolic conservation laws, but may fail to capture composite structures in non-convex flux cases. A hybrid WENO scheme has been proposed to resolve these issues by identifying troubled-cells, using first-order monotone modifications in those cells, and employing standard WENO reconstruction in non-troubled cells. This approach allows for high accuracy while ensuring correct resolution of composite structures, making it a useful tool for a variety of applications.
  • Classification : 65M06, 65M08, Hyperbolic conservation laws, WENO reconstruction, Finite difference methods
  • Author(s) :
    • Asha Kisan Dond (Indian Institute of Science Education and Research Thiruvananthapuram)
    • Rakesh Kumar (Indian Institute of Science Education and Research Thiruvananthapuram, India)

[02560] A Composite Adaptive Finite Point Method for 2D Burgers' Equation

  • Session Date & Time : 2E (Aug.22, 17:40-19:20)
  • Type : Contributed Talk
  • Abstract : This paper focuses on solving the unsteady 2D Burgers equation. We present a minimal machinery algorithm based on an operator splitting technique into different temporal levels in conjunction with an adaptive finite point method (AFPM). The advisability of the AFPM is that it efficiently adjusts itself to fit into the local properties of the exact solution, and its user-friendliness makes it easy to implement and cost-effective. Mathematical estimates like stability, consistency, and convergence analysis support the presented method.
  • Classification : 65M06, 65M12
  • Author(s) :
    • Ashish Awasthi (National Institute of Technology Calicut)
    • Sreelakshmi A (National Institute of Technology Calicut)
    • Shyaman V P (National Institute of Technology Calicut)

[01004] An Error Estimate for an Implicit-Upwind Finite Volume Scheme for Boussinesq Model

  • Session Date & Time : 2E (Aug.22, 17:40-19:20)
  • Type : Contributed Talk
  • Abstract : This study contains an error estimate for a Finite Volume Method-based Implicit-Upwind scheme for the d-dimensional(d=2 or 3) Boussinesq Model, which describes several buoyancy-driven Hydrodynamic phenomena such as natural-convection in a cavity and Marsigli Flow. For each time level, the L2- norms of the error for the temperature and velocity components are found to be of order (h + k), where h is the spatial grid size and k is the time step size.
  • Classification : 65M08, 65M15, 65N08, 65N15
  • Author(s) :
    • Chitranjan Pandey (Indian Institute of Technology Kanpur, India)
    • B.V. Rathish Kumar (Indian Institute of Technology Kanpur, India)

[01142] Thermodynamically Consistent Finite Volume Schemes for Electrolyte Simulations

  • Session Date & Time : 2E (Aug.22, 17:40-19:20)
  • Type : Contributed Talk
  • Abstract : In order to account for finite ion sizes and solvation effects, the classical Nernst-Planck-Poisson system describing ion transport in electrolytes needs to be enhanced by cross-diffusion - like terms. For this situation, we present a space discretization scheme which adapts the classical Scharfetter-Gummel exponential fitting upwind flux for the Voronoi box finite volume method. Numerical examples use the Julia package VoronoiFVM.jl which takes advantage of automatic differentiation to handle the strong nonlinearities of the system.
  • Classification : 65M08, 65M12, 78A57, 65N08, 35Q81
  • Author(s) :
    • Jürgen Fuhrmann (Weierstrass Institute for Applied Analysis and Stochastics)
    • Benoit Gaudeul (Univ. Paris Saclay)