Computational Methods in Applied Mathematics

Published by De Gruyter

Volume 24 Issue 1

Issue of Computational Methods in Applied Mathematics

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Contents
Journal Overview

Publicly Available January 1, 2024

Frontmatter

Page range: i-iv

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July 12, 2023

Approximate Deconvolution with Correction – A High Fidelity Model for Magnetohydrodynamic Flows at High Reynolds and Magnetic Reynolds Numbers

Yasasya Batugedara, Alexander E. Labovsky

Page range: 1-20

Abstract

We propose a model for magnetohydrodynamic flows at high Reynolds and magnetic Reynolds numbers. The system is written in the Elsässer variables so that the decoupling method of [C. Trenchea, Unconditional stability of a partitioned IMEX method for magnetohydrodynamic flows, Appl. Math. Lett. 27 (2014), 97–100] can be used. This decoupling method is only first-order accurate, so the proposed model aims at improving the temporal accuracy (from first to second order), as well as reducing the modeling error of the existing turbulence model. This is done in the framework of the recently developed LES-C turbulence models [A. E. Labovsky, Approximate deconvolution with correction: A member of a new class of models for high Reynolds number flows, SIAM J. Numer. Anal. 58 (2020), 5, 3068–3090]. We show the model to be unconditionally stable and numerically verify its superiority over its most natural competitor.

July 8, 2023

Well-Posedness and Convergence Analysis of PML Method for Time-Dependent Acoustic Scattering Problems Over a Locally Rough Surface

Hongxia Guo, Guanghui Hu

Page range: 21-48

Abstract

We aim to analyze and calculate time-dependent acoustic wave scattering by a bounded obstacle and a locally perturbed non-selfintersecting curve. The scattering problem is equivalently reformulated as an initial-boundary value problem of the wave equation in a truncated bounded domain through a well-defined transparent boundary condition. Well-posedness and stability of the reduced problem are established. Numerically, we adopt the perfect matched layer (PML) scheme for simulating the propagation of perturbed waves. By designing a special absorbing medium in a semi-circular PML, we show the well-posedness and stability of the truncated initial-boundary value problem. Finally, we prove that the PML solution converges exponentially to the exact solution in the physical domain. Numerical results are reported to verify the exponential convergence with respect to absorbing medium parameters and thickness of the PML.

April 27, 2023

A New Immersed Finite Element Method for Two-Phase Stokes Problems Having Discontinuous Pressure

Gwanghyun Jo, Do Young Kwak

Page range: 49-58

Abstract

In this paper, we develop a new immersed finite element method (IFEM) for two-phase incompressible Stokes flows. We allow the interface to cut the finite elements. On the noninterface element, the standard Crouzeix–Raviart element and the P 0 {P_{0}} element pair is used. On the interface element, the basis functions developed for scalar interface problems (Kwak et al., An analysis of a broken P 1 {P_{1}} -nonconforming finite element method for interface problems, SIAM J. Numer. Anal. (2010)) are modified in such a way that the coupling between the velocity and pressure variable is different. There are two kinds of basis functions. The first kind of basis satisfies the Laplace–Young condition under the assumption of the continuity of the pressure variable. In the second kind, the velocity is of bubble type and is coupled with the discontinuous pressure, still satisfying the Laplace–Young condition. We remark that in the second kind the pressure variable has two degrees of freedom on each interface element. Therefore, our methods can handle the discontinuous pressure case. Numerical results including the case of the discontinuous pressure variable are provided. We see optimal convergence orders for all examples.

July 12, 2023

Simultaneous Recovery of Two Time-Dependent Coefficients in a Multi-Term Time-Fractional Diffusion Equation

Wenjun Ma, Liangliang Sun

Page range: 59-83

Abstract

This paper deals with an inverse problem on simultaneously determining a time-dependent potential term and a time source function from two-point measured data in a multi-term time-fractional diffusion equation. First we study the existence, uniqueness and some regularities of the solution for the direct problem by using the fixed point theorem. Then a nice conditional stability estimate of inversion coefficients problem is obtained based on the regularity of the solution to the direct problem and a fine property of the Caputo fractional derivative. In addition, the ill-posedness of the inverse problem is illustrated and we transfer the inverse problem into a variational problem. Moreover, the existence and convergence of the minimizer for the variational problem are given. Finally, we use a modified Levenberg–Marquardt method to reconstruct numerically the approximate functions of two unknown time-dependent coefficients effectively. Numerical experiments for three examples in one- and two-dimensional cases are provided to show the validity and robustness of the proposed method.

May 31, 2023

𝐻²-Conformal Approximation of Miura Surfaces

Frédéric Marazzato

Page range: 85-100

Abstract

The Miura ori is a very classical origami pattern used in numerous applications in engineering. A study of the shapes that surfaces using this pattern can assume is still lacking. A constrained nonlinear partial differential equation (PDE) that models the possible shapes that a periodic Miura tessellation can take in the homogenization limit has been established recently and solved only in specific cases. In this paper, the existence and uniqueness of a solution to the unconstrained PDE is proved for general Dirichlet boundary conditions. Then an H 2 H^{2} -conforming discretization is introduced to approximate the solution of the PDE coupled to a Newton method to solve the associated discrete problem. A convergence proof for the method is given as well as a convergence rate. Finally, numerical experiments show the robustness of the method and that nontrivial shapes can be achieved using periodic Miura tessellations.

May 23, 2023

A Second-Order Difference Scheme for Generalized Time-Fractional Diffusion Equation with Smooth Solutions

Aslanbek Khibiev, Anatoly Alikhanov, Chengming Huang

Page range: 101-117

Abstract

In the current work, we build a difference analog of the Caputo fractional derivative with generalized memory kernel ( 𝜇 L2-1 𝜎 formula). The fundamental features of this difference operator are studied, and on its ground, some difference schemes generating approximations of the second order in time for the generalized time-fractional diffusion equation with variable coefficients are worked out. We have proved stability and convergence of the given schemes in the grid L 2 L_{2} -norm with the rate equal to the order of the approximation error. The achieved results are supported by the numerical computations performed for some test problems.

Publicly Available May 6, 2023

Volume Integral Equations and Single-Trace Formulations for Acoustic Wave Scattering in an Inhomogeneous Medium

Ignacio Labarca, Ralf Hiptmair

Page range: 119-139

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Abstract

We study frequency domain acoustic scattering at a bounded, penetrable, and inhomogeneous obstacle Ω − ⊂ R d \Omega^{-}\subset\mathbb{R}^{d} , d = 2 , 3 d=2,3 . By defining constant reference coefficients, a representation formula for the pressure field is derived. It contains a volume integral operator, related to the one in the Lippmann–Schwinger equation. Besides, it features integral operators defined on ∂ Ω − \partial\Omega^{-} and closely related to boundary integral equations of single-trace formulations (STF) for transmission problems with piecewise constant coefficients. We show well-posedness of the continuous variational formulation and asymptotic convergence of Galerkin discretizations. Numerical experiments in 2D validate our expected convergence rates.

July 12, 2023

An Adaptive Two-Grid Solver for DPG Formulation of Compressible Navier–Stokes Equations in 3D

Waldemar Rachowicz, Witold Cecot, Adam Zdunek

Page range: 141-172

Abstract

We present an overlapping domain decomposition iterative solver for linear systems resulting from the discretization of compressible viscous flows with the Discontinuous Petrov–Galerkin (DPG) method in three dimensions. It is a two-grid solver utilizing the solution on the auxiliary coarse grid and the standard block-Jacobi iteration on patches of elements defined by supports of the coarse mesh base shape functions. The simple iteration defined in this way is used as a preconditioner for the conjugate gradient procedure. Theoretical analysis indicates that the condition number of the preconditioned system should be independent of the actual finite element mesh and the auxiliary coarse mesh, provided that they are quasiuniform. Numerical tests confirm this result. Moreover, they show that presence of strongly flattened or elongated elements does not slow the convergence. The finite element mesh is subject to adaptivity, i.e. dividing the elements with large errors until a required accuracy is reached. The auxiliary coarse mesh is adjusting to the nonuniform actual mesh.

June 15, 2023

An Efficient Discretization Scheme for Solving Nonlinear Ill-Posed Problems

M. P. Rajan, Jaise Jose

Page range: 173-184

Abstract

Information based complexity analysis in computing the solution of various practical problems is of great importance in recent years. The amount of discrete information required to compute the solution plays an important role in the computational complexity of the problem. Although this approach has been applied successfully for linear problems, no effort has been made in literature to apply it to nonlinear problems. This article addresses this problem by considering an efficient discretization scheme to discretize nonlinear ill-posed problems. We apply the discretization scheme in the context of a simplified Gauss–Newton iterative method and show that our scheme requires only less amount of information for computing the solution. The convergence analysis and error estimates are derived. Numerical examples are provided to illustrate the fact that the scheme can be implemented successfully. The theoretical and numerical study asserts that the scheme can be employed to nonlinear problems.

Publicly Available May 31, 2023

Tensor-Product Space-Time Goal-Oriented Error Control and Adaptivity With Partition-of-Unity Dual-Weighted Residuals for Nonstationary Flow Problems

Julian Roth, Jan Philipp Thiele, Uwe Köcher, Thomas Wick

Page range: 185-214

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Abstract

In this work, the dual-weighted residual method is applied to a space-time formulation of nonstationary Stokes and Navier–Stokes flow. Tensor-product space-time finite elements are being used to discretize the variational formulation with discontinuous Galerkin finite elements in time and inf-sup stable Taylor–Hood finite element pairs in space. To estimate the error in a quantity of interest and drive adaptive refinement in time and space, we demonstrate how the dual-weighted residual method for incompressible flow can be extended to a partition-of-unity based error localization. We substantiate our methodology on 2D benchmark problems from computational fluid mechanics.

June 6, 2023

Identification of an Inverse Source Problem in a Fractional Partial Differential Equation Based on Sinc-Galerkin Method and TSVD Regularization

Ali Safaie, Amir Hossein Salehi Shayegan, Mohammad Shahriari

Page range: 215-237

Abstract

In this paper, using Sinc-Galerkin method and TSVD regularization, an approximation of the quasi-solution to an inverse source problem is obtained. To do so, the solution of direct problem is obtained by the Sinc-Galerkin method, and this solution is applied in a least squares cost functional. Then, to obtain an approximation of the quasi-solution, we minimize the cost functional by TSVD regularization. Error analysis and convergence of the proposed method are investigated. In addition, at the end, four numerical examples are given in details to show the efficiency and accuracy of the proposed method.

June 15, 2023

A Formulation for a Nonlinear Axisymmetric Magneto-Heat Coupling Problem with an Unknown Nonlocal Boundary Condition

Ran Wang, Huai Zhang, Tong Kang

Page range: 239-264

Abstract

This paper investigates a nonlinear axisymmetric magneto-heat coupling problem described by the quasi-static Maxwell’s equations and a heat equation. The coupling between them is provided through the temperature-dependent electric conductivity. The behavior of the material is defined by an anhysteretic 𝑯-𝑩 curve. The magnetic flux across a meridian section of the medium gives rise to the magnetic field equation with the unknown nonlocal boundary condition. We present a variational formulation for this coupling problem and prove its solvability in terms of the Rothe method. The nonlinearity is handled by the theory of monotone operators. We also suggest a discrete decoupled scheme to solve this problem by employing the finite element method and show some numerical results in the final section.

June 15, 2023

Landweber Iterative Method for an Inverse Source Problem of Time-Space Fractional Diffusion-Wave Equation

Fan Yang, Yan Zhang, Xiao-Xiao Li

Page range: 265-278

Abstract

In this paper, we apply a Landweber iterative regularization method to determine a space-dependent source for a time-space fractional diffusion-wave equation from the final measurement. In general, this problem is ill-posed, and a Landweber iterative regularization method is used to obtain the regularization solution. Under the a priori parameter choice rule and the a posteriori parameter choice rule, we give the error estimates between the regularization solution and the exact solution, respectively. Some numerical results in the one-dimensional and two-dimensional cases show the utility of the used regularization method.

About this journal

Objective
The highly selective international mathematical journal Computational Methods in Applied Mathematics (CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs.

CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics.

Topics

Numerical and computational mathematics
Partial differential equations
Applied mathematics

Article formats
Original research articles

Proposals for special issues of CMAM are considered.
Note that for special issue proposals not only an exciting topic within the scientific scope of the journal is required, but also the Guest Editors need to have an outstanding worldwide reputation in their field.

Volume 24 Issue 1

Issue of Computational Methods in Applied Mathematics

Frontmatter

Approximate Deconvolution with Correction – A High Fidelity Model for Magnetohydrodynamic Flows at High Reynolds and Magnetic Reynolds Numbers

Abstract

Well-Posedness and Convergence Analysis of PML Method for Time-Dependent Acoustic Scattering Problems Over a Locally Rough Surface

Abstract

A New Immersed Finite Element Method for Two-Phase Stokes Problems Having Discontinuous Pressure

Abstract

Simultaneous Recovery of Two Time-Dependent Coefficients in a Multi-Term Time-Fractional Diffusion Equation

Abstract

𝐻2-Conformal Approximation of Miura Surfaces

Abstract

A Second-Order Difference Scheme for Generalized Time-Fractional Diffusion Equation with Smooth Solutions

Abstract

Volume Integral Equations and Single-Trace Formulations for Acoustic Wave Scattering in an Inhomogeneous Medium

Abstract

An Adaptive Two-Grid Solver for DPG Formulation of Compressible Navier–Stokes Equations in 3D

Abstract

An Efficient Discretization Scheme for Solving Nonlinear Ill-Posed Problems

Abstract

Tensor-Product Space-Time Goal-Oriented Error Control and Adaptivity With Partition-of-Unity Dual-Weighted Residuals for Nonstationary Flow Problems

Abstract

Identification of an Inverse Source Problem in a Fractional Partial Differential Equation Based on Sinc-Galerkin Method and TSVD Regularization

Abstract

A Formulation for a Nonlinear Axisymmetric Magneto-Heat Coupling Problem with an Unknown Nonlocal Boundary Condition

Abstract

Landweber Iterative Method for an Inverse Source Problem of Time-Space Fractional Diffusion-Wave Equation

Abstract

𝐻²-Conformal Approximation of Miura Surfaces