Abstract : This mini-symposium for ICIAM2023 concerns different important topics such as mathematical modeling, theoretical analysis, and numerical methods. An important goal of this workshop is to foster collaboration between mathematicians, computational scientists, and engineers.
Applications include classic interface problems, Newtonian and non-Newtonian fluids, fluid and porous media, or viscoelastic, or poroelastic media couplings, finite element, finite volume, and finite differences or other numerical methods. Wellposedness of mathematical models and so on. The nature of this workshop will be mathematics centered with multi-disciplinary and multi-physics applications.
[02815] High Order Compact Finite Difference Schemes for Helmholtz Interface Problem
Format : Talk at Waseda University
Author(s) :
Bin Han (University of Alberta)
Qiwei Feng (University of Alberta)
Michelle Michelle (Purdue University)
Yau Shu Wong (University of Alberta)
Abstract : The Helmholtz equation is numerically challenging to solve, due to highly oscillating solutions and ill-conditioned huge matrices. Introducing Dirac-Assisted-Tree DAT method and high-order compact FDMs, we can handle 1D-heterogeneous and special 2D-Helmholtz interface problem with large wavenumbers by only solving small linear systems. We present 5th-order compact FDMs for 2D-Helmholtz interface problem with discontinuous wavenumbers and reduced pollution effect. Numerical experiments demonstrate effectiveness and superior performance of our proposed methods for Helmholtz interface problem.
[03418] Cubic Hermite splines plus correction terms: a way of adaption to the presence of singularities
Format : Talk at Waseda University
Author(s) :
Juan Ruiz-Alvarez (Universidad Politecnica de Cartagena)
Zhilin Li (North Carolina State University)
Sergio Amat (Universidad Politécnica de Cartagena)
Juan Carlos Trillo (Universidad Politécnica de Cartagena)
Concepción Solano (Universidad Politécnica de Cartagena)
Abstract : Hermite interpolation is classically used to reconstruct smooth data when the function and its derivatives are available at certain nodes. If derivatives are not available, it is easy to set a system of equations imposing some regularity conditions at the data nodes in order to obtain them. This process leads to the construction of a Hermite spline. The problem of the described Hermite splines is that the accuracy is lost if the data contains singularities. The consequence is the appearance of oscillations, if there is a jump discontinuity in the function, that globally affects the accuracy of the spline, or the smearing of singularities, if the discontinuities are in the derivatives of the function. This work is devoted to the construction and analysis of a new technique that allows for the computation of accurate derivatives of a function close to singularities using a Hermite spline. The idea is inspired in the immersed interface method (IIM) and aims to correct the system of equations of the spline in order to attain the desired accuracy even close to the singularities. Once we have computed the derivatives with enough accuracy, a correction term is added to the Hermite spline in the intervals that contain a singularity. The aim is to reconstruct piecewise smooth functions with $O(h^4)$ accuracy even close to the singularities. The process of adaption requires some knowledge about the position of the singularity and the values of the function and its derivatives at the singularity. The whole process can be used as a post-processing, where a correction term is added to the classical cubic Hermite spline. Proofs for the accuracy and regularity of the corrected spline and its derivatives are given. We also analyse the mechanism that eliminates the Gibbs phenomenon close to jump discontinuities in the function. The numerical experiments presented confirm the theoretical results obtained.
[02836] Difference Finite Element Method for 3D Steady Navier-Stokes Equations
Format : Talk at Waseda University
Author(s) :
Xinlong Feng (Xinjiang University)
Abstract : In this work, a difference finite element (DFE) method for the 3D steady Navier–Stokes (N–S) equations is presented. This new method consists of transmitting the FE solution of 3D steady N–S equations into a series of the FE solutions of 2D steady Oseen iterative equations, which are solved by using the finite element pair (P1b,P1b,P1)×P1 satisfying the discrete inf-sup condition in a 2D domain ω . In addition, we use finite element pair ((P1b,P1b,P1)×P1)×(P1×P0) to solve 3D steady Oseen iterative equations, where the pair satisfies the discrete inf-sup condition in a 3D domain Ω under the quasi-uniform mesh condition. To overcome the difficulty of nonlinearity, we apply the Oseen iterative method and present the weak formulation of the DFE method for solving 3D steady Oseen iterative equations. Moreover, we provide the existence and uniqueness of the DFE solutions of 3D steady Oseen iterative equations and deduce the first order convergence with respect to the discrete step parameter of the DFE solutions to the exact solution of 3D steady N–S equations. Finally, numerical tests are presented to show the accuracy and effectiveness of the proposed method.
[03698] Finite difference method on staggered grids for Stokes-Biot problems
Format : Online Talk on Zoom
Author(s) :
Hongxing Rui (Shandong University)
Abstract : In this talk, we will present a looking-free finite difference method based on staggered grids for coupled Stokes-Biot problems. The model problems are used to describe Stokes fluid coupled with a poroelastic flow with a interfece. The construction of the finite difference schemes, analysis for the existence and uniqueness of the approximation solutions, superconvergence will be presented. Numerical experiments are presented to confirm the theoretical results. Then we will present a semi-decoupled scheme for the Stokes-Biot system, where the displacement of structure is split from the whole system using the time-lagging scheme. We will also present some ongoing works on coupled problem briefly.
Abstract : \title{\bf Value function approximation of PDEs}
\author{Kazufumi Ito\thanks{Department of Mathematics, North Carolina State University, USA}
\noindent {\bf Abstract} In this paper we discuss a value function approximation of a general class
of nonlinear system of parabolic equations. Our approach is based on the backward stochastic
differential equations of nonlinear expectation. The approach uses the discrete time dynamic programing
formulation of the value function update. It results in an operator splitting of
the diffusion term and a semi-implicit method for the nonlinear hyperbolic term.
It is very easy to implemented and provides an accurate value function approximation.
We apply the method several applications including elliptic interface problems,
conservation laws and Navier-Stokes equations.
We analyze the stability and convergence of the proposed method.
Numerical results are presented to demonstrate the applicability
[03699] A hybrid asymptotic and augmented compact FVM for degenerate interface problem with extreme conditions
Format : Talk at Waseda University
Author(s) :
Zhiyue Zhang (Nanjing Normal University)
Abstract : An accurate and efficient numerical method has been proposed for degenerate interface problem with extreme conditions such as very big jump ratio, coefficient blow-up and geometric singularity interface . The scheme combines Puiseux series asymptotic technique with augmented fourth order compact finite volume method for the problem. Error estimates are obtained. Numerical examples confirm the theoretical analysis and efficiency of the method. We also apply this method for solving time dependent problems and 2D problems.
[03531] A fast front-tracking approach for a temporal multiscale blood flow problem
Format : Talk at Waseda University
Author(s) :
Ping Lin (University of Dundee)
Zhenlin Guo (Beijing Computational Science Research Center)
Abstract : We consider a blood flow problem (fast system) coupled with a slow plaque growth with memory effect (slow system) at the artery wall. We construct an auxiliary temporal periodic problem and an effective time-average equation to
approximate the original problem and analyze the approximation error of the corresponding PDE system, where the
front-tracking technique is used to update the moving boundary. An effective multiscale method is then
designed and its approximation error is analysed.
[03479] An Energy Stable Immersed Boundary Method for Deformable Membrane Problem with Non-uniform Density and Viscosity
Format : Online Talk on Zoom
Author(s) :
Dongdong He (The Chinese University of Hong Kong, Shenzhen)
Qinghe Wang (The Chinese University of Hong Kong, Shenzhen)
Mingyang Pan (Hebei University of Technology)
Yu-Hau Tseng (Kaohsiung University)
Abstract : Membrane problems commonly encountered in engineering and biological applications involve large deformations and complex configurations. Immersed boundary method, formulated by the fluid equations in which the fluid-structure interaction is described in terms of the Dirac function, is one of the most powerful tools to simulate such problems. However, the IB method suffers from severe time step restrictions to maintain stability if the discretization lacks conservation of energy, especially for two-phase flows. In this paper, we develop an energy stable IB method for solving deformable membrane problems with non-uniform density and viscosity. Unlike the classic IB formulation, the evolution of membrane, including elastic tension and bending force, is controlled by its tangent angle and arc length. After minor modifications, it is shown that the model satisfies the continuous energy law. Thus, for the reformulated model, we proposed an implicit unconditionally energy stable scheme, where the energy of the scheme is proved to be dissipative. The resultant system is solved iteratively and the numerical results show that the proposed scheme is energy stable and capable of predicting the dynamics of extensible and inextensible interface problems with non-uniform density and viscosity.
