International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
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Please note that all times are shown in the time zone of the conference. The current conference time is: 10th Dec 2022, 10:28:57am CET
Z2a: DG, VEM, structure preservation
12:00pm - 12:20pm
A TREFFTZ METHOD WITH RECONSTRUCTION OF THE NORMAL DERIVATIVE APPLIED TO ELLIPTIC EQUATIONS
1Sorbonne University, France; 2Saint Joseph University, Lebanon
There are many classical numerical methods for solving boundary value problems on general domains. The Trefftz method is an approximation method for solving linear boundary value problems arising in applied mathematics and engineering sciences. This method consists to approximate the exact solution through a linear combination of trial functions satisfying exactly the governing differential equation. One of the advantages of this method is that the number of trial functions per cell is O(m), asymptotically much less than the quadratic estimate O(m^2) for finite element and discontinuous Galerkin approximations.
For a Laplace model equation, we present a high order Trefftz method with quadrature formula for calculation of normal derivative at interfaces. We introduce a discrete variational formulation and study the existence and uniqueness of the discrete solution. A priori error estimate is then established and finally, several numerical experiments are shown.
12:20pm - 12:40pm
Explicit high-order unconditionally structure-preserving schemes for the conservative Allen-Cahn equations
1Department of Mathematics, National University of Defense Technology, People's Republic of China; 2Division of Mathematical Sciences, Nanyang Technological University, 637371, Singapore
Comparing with the well-known classical Allen-Cahn equation, the modified Allen-Cahn equation which is equipped with a nonlocal Lagrange multiplier or a local-nonlocal Lagrange multiplier, enforces the mass conservation for modeling phase transitions. In this work, a class of up to third-order explicit structure-preserving schemes is proposed for solving these two modified conservative Allen-Cahn equations. Based on the second-order finite difference space discretization, we investigate the newly developed improved stabilized integrating factor Runge-Kutta (isIFRK) schemes for the stabilizing reformulations of the conservative Allen-Cahn equations. We prove that the original stabilized integrating factor Runge-Kutta schemes fail to preserve the mass conservation law when the stabilizing constant $\kappa > 0$ and the initial mass not equals zero, while the isIFRK schemes not only preserve the maximum-principle unconditionally, but also conserve the mass to machine accuracy without any restriction on the time step size. Convergence and energy stability of the proposed schemes are also presented. Furthermore, a series of numerical experiments validate that each reformulation of the conservative Allen-Cahn equations has it own advantage, and the isIFRK schemes can reach the expected high-order accuracy, conserve the mass and preserve the maximum-principle unconditionally.
12:40pm - 1:00pm
Virtual element methods for nonlinear problems
Indian Institute of Space Science and Technology Thiruvananthapuram Kerala, India
In this talk, we discuss and analyze new conforming virtual element methods (VEMs) for the approximation of nonlinear problems on convex polygonal meshes in two spatial dimension. The spatial discretization is based on polynomial and suitable nonpolynomial functions. The basis functions are not evaluated explicitly but the mass and stiffness matrices are computed only using the degrees of freedom. This is carried out with the help of suitable projection operators. The discrete formulation of both the proposed schemes is discussed in detail, and the unique solvability of the resulted schemes is discussed. A priori error estimates for the proposed schemes in $H^1$ and $L^2$ norms are derived under the assumption that the source term $f$ is Lipschitz continuous. Some numerical experiments are conducted to illustrate the performance of the proposed scheme and to confirm the theoretical convergence rates.
1:00pm - 1:20pm
Unified framework for conservative discontinuous Galerkin methods for nonlinear Schrödinger equations
1University of Puerto Rico; 2University of Pavia, Italy
In this talk we present a family of fully discrete methods preserving the discrete version of two important physical invariants for the nonlinear Schrödinger (NLS) equation with generalized potential and their extension to N strongly coupled nonlinear Schrödinger (N-CNLS) systems.
These methods combine a class of symmetric discontinuous Galerkin (dG) methods as spatial discretization and modified Crank-Nicolson time marching schemes. For N-CNLS systems, in order to avoid solving a global nonlinear system, involving all the components of the vector field at each time step, a conservative nonlinear splitting method is proposed. Conservation of the mass for each component and total energy is formally proven for the semi-discrete and fully-discrete methods. Conservation and accuracy of the proposed methods are numerically validated on a series of benchmark problems. In particular, for the minimal dissipation version of the Local Discontinuous Galerkin (md-LDG) method; using a special projector operator, the approximated initial energy of the system exhibits a convergence with order O(h^(2p+2)) when polynomial approximations of degree p are used.
1:20pm - 1:40pm
Invariant Energy Quadratiation and Scalar Auxiliary Variable methods applied to piano strings modeling
Makutu team, INRIA - LMAP Univ. Pau, France
This work consists in modeling each part of the piano from the hammer to the sound radiation in order to compute the sound directly from the pianist’s performance. Each part is modelled with a set of equations along with numerical methods in space and time specifically designed according to the corresponding mathematical properties. All these schemes are then coupled to simulate the whole piano. Techniques based on energy balance allow to prove the stability of models and associated numerical schemes, and ensure global stability in the coupled situation.
The numerous nonlinearities of the models are difficult to solve. We will focus here on the distributed geometric nonlinearity of the piano string.
A Galerkin variationnal method, more precisely a high order spectral finite element method, is used for spatial discretization.
“Discrete gradient” time schemes lead to implicit and nonlinear discretizations which implies to solve a nonlinear problem at every time step, with an iterative Newton scheme for example. Using Invarient Energy Quadratization (IEQ) and Scalar Auxiliary Variable (SAV) methods to reformulate the equations leads to linearly implicit schemes with a much faster computation.
We willl present an adaptation of these techniques for the piano string by introducing auxiliary variables in the variationnal formulation of the equations and we will analyse their efficiency and accuracy. A bounded function linked to the auxiliary variables allows to prove quadratic convergence for both schemes.
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