MS12: Recent advances in numerical algorithms for partial differential equations and applications


Session A: Oct.1, 10:40am-12:00pm, Classroom Building 112
Session B: Oct.1, 3:10pm-4:30pm, Classroom Building 112
Session C: Oct.1, 5:00pm-6:20pm, Classroom Building 112
Session D: Oct.2, 10:30am-11:50am, Classroom Building 112

10:40am-11:00am (Oct 1)
CLB 112

Tom Lewis,
The University of North Carolina at Greensboro

Convergent finite difference methods with higher order local truncation errors for stationary Hamilton-Jacobi equations


A new non-monotone finite difference (FD) method for approximating viscosity solutions of stationary Hamilton-Jacobi problems with Dirichlet boundary conditions will be discussed. The new FD method has local truncation errors that are above the first order Godunov barrier for monotone methods. The method uses a stabilization term called a numerical moment to ensure that the proposed scheme is admissible, stable, and convergent. Numerical tests will be provided that compare the accuracy of the proposed scheme to that of the Lax-Friedrich's method.

11:00am-11:20am (Oct 1)
CLB 112
Amanda Diegel,
Mississippi State University

Continuous data assimilation and long-time accuracy in a C0-IP Method for the Cahn-Hilliard Equation
  We discuss a numerical approximation method for the Cahn-Hilliard equation that incorporates continuous data assimilation in order to achieve long time accuracy. The method uses a C0 interior penalty spatial discretization of the fourth order Cahn-Hilliard equation, together with a semi-implicit temporal discretization. The method is long time stable and long time accurate, for arbitrarily inaccurate initial conditions, provided enough data measurements are incorporated into the simulation. Numerical experiments illustrate the effectiveness of the method on a benchmark test problem.
11:20am-11:40am (Oct 1)
CLB 112
John Carter,
Missouri University of Science and Technology
SAV Ensemble Algorithms for the magnetohydrodynamics equations.
  We develop two linear, second-order accurate, unconditionally stable ensemble methods with shared coefficient matrix across different realizations and time steps for the magnetohydrodynamics equations. The viscous terms are treated by a standard perturbative discretization. We employ the Generalized Positive Auxiliary Variable method to discretize nonlinear terms, resulting in linearity of the algebra equation for the scalar variable, provable positivity of the scalar variable, and flexibility in handling complex boundary conditions. Artificial viscosity stabilization that modifies the kinetic energy is introduced to improve accuracy of the GPAV ensemble methods.
11:40am-12:00pm (Oct 1)
CLB 112
Qifan Chen,
The Ohio State University

Runge-Kutta discontinuous Galerkin methods with compact stencils for hyperbolic conservation laws
  In this talk, we develop a new type of Runge-Kutta (RK) discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws. Compared with the standard RKDG methods, the new methods feature improved compactness and allow simple boundary treatment. The convergence to weak solution and the accuracy of the numerical solutions are studied. Their connections with the Lax-Wendroff DG schemes and the ADER DG schemes are also investigated. Numerical examples are given to confirm that the new RKDG schemes are as accurate as standard RKDG methods, while being more compact and cost-effective, for a wide range of problems including two-dimensional Euler systems of compressible gas dynamics.
3:10pm-3:30pm (Oct 1)
CLB 112
Ari Stern,
Washington University in St. Louis

A nonconforming primal hybrid finite element method for the two-dimensional vector Laplacian

We introduce a nonconforming hybrid finite element method for the two-dimensional vector Laplacian, based on a primal variational principle for which conforming methods are known to be inconsistent. Consistency is ensured using penalty terms similar to those used to stabilize hybridizable discontinuous Galerkin (HDG) methods, with a carefully chosen penalty parameter due to Brenner, Li, and Sung [Math. Comp., 76 (2007), pp. 573-595]. Our method accommodates elements of arbitrarily high order and, like HDG methods, it may be implemented efficiently using static condensation. The lowest-order case recovers the P1-nonconforming method of Brenner, Cui, Li, and Sung [Numer. Math., 109 (2008), pp. 509-533], and we show that higher-order convergence is achieved under appropriate regularity assumptions. The analysis makes novel use of a family of weighted Sobolev spaces, due to Kondrat'ev, for domains admitting corner singularities.

Joint work with Mary Barker and Shuhao Cao.

3:30pm-3:50pm (Oct 1)
CLB 112
Qian Zhang,
Michigan Technological University

Spurious solutions for high-order curl problems


In this talk, we investigate numerical solutions of high-order curl problems with various formulations and finite elements. We show that several classical conforming finite elements lead to spurious solutions, while mixed formulations with finite elements in complexes solve the problems correctly. We explain the numerical results by clarifying the cohomological structures in high-order curl problems and relate the partial differential equations to the Hodge–Laplacian boundary problems of the grad curl complexes.

3:50pm-4:10pm (Oct 1)
CLB 112
Qiao Zhuang,
Worcester Polytechnic Institute

An immersed Crouzeix-Raviart finite element method for Navier-Stokes equations with moving interfaces
  In this talk, we introduce a Cartesian-mesh finite element method for solving Navier- Stokes interface problems with moving interfaces. The spatial discretization uses the immersed Crouzeix-Raviart nonconforming finite element introduced by Jones and Zhang (JCAM, 2021). A backward Euler full-discrete scheme is developed which embeds Newton’s iteration to treat the nonlinear convective term. The proposed IFE method does not require any stabilization terms while maintaining its convergence in optimal order. Numerical experiments with various interface shapes and jump coefficients are provided to demonstrate the accuracy of the proposed method. Numerical results indicate the optimal order of convergence of the IFE method.

4:10pm-4:30pm (Oct 1)
CLB 112
Yuan Chen,
The Ohio State University

A high-order Immersed C0 interior penalty method for biharmonic interface problems
  In this talk, an immersed C0 interior penalty method is proposed to solve biharmonic interface problems on unfitted mesh. The immersed P2 and P3 finite element spaces are constructed to match biharmonic interface conditions in a least-squares sense. Basic properties of these new spaces such as unisolvence and partition of unity are analyzed. The new proposed spaces are also used in a symmetric C0 interior penalty scheme to solve the biharmonic interface problems. The well-posedness of discrete solution is also proved. Extensive numerical experiments show optimal convergence of proposed method in L2, H1 and H2 norms. This is a joint work with Dr. Xu Zhang.
5:00pm-5:20pm (Oct 1)
CLB 112
Tao Lin,
Virginia Tech

Solving interface problems by immersed spline functions
  We consider a group of interface problems derived from the desire to solve the Stefan problem. We try to solve these interface problems on meshes that are independent of the interface location. The underlying differential equations are discretized by the collocation with spline functions defined on an interface independent mesh. For time-dependent interface problems, the method of line or other popular discretization approaches for the time variable can be adopted. Numerical examples will be presented to demonstrate features of this method.
5:20pm-5:40pm (Oct 1)
CLB 112
Zheng Sun,
University of Alabama

On a numerical artifact of solving shallow water equations with a discontinuous bottom
  In this talk, we discuss a numerical artifact of solving the nonlinear shallow water equations with discontinuous bottom topography. For a few first-order schemes, the numerical solution will form a spurious spike in the momentum, which should not exist in the exact solution. The height of the spike cannot be reduced by the mesh refinement. In many problems, this numerical artifact may cause the wrong convergence, which means that the limit of the numerical solution is not a weak solution of the shallow water equations. To explain the formation of the spurious spike, we perform a convergence analysis of the numerical methods. It is shown that the spurious spike is caused by the numerical viscosity at the discontinuous bottom and its height is proportional to the viscosity constant in the numerical flux. Furthermore, by adopting appropriate modifications at the bottom discontinuity, we show that this numerical artifact can be removed in many cases. For various of numerical tests with nontransonic Riemann solutions, the modified scheme is able to retrieve the correct convergence.
5:40pm-6:00pm (Oct 1)
CLB 112
Qingguang Guan,
University of Southern Mississippi

Modeling Calcium Dynamics in Neurons with Endoplasmic Reticulum: Well-Posedness and Numerical Methods
  Calcium dynamics in neurons containing an endoplasmic reticulum are governed by diffusion-reaction equations on interface-separated domains. Interface conditions are typically described by systems of ordinary differential equations that provide fluxes across the interfaces. For the model with ODE-flux boundary condition, we prove the existence, uniqueness, and boundedness of the solution. We develop a stable high-order multi-step scheme to overcome the instability and low accuracy of the one-step implicit-explicit method. Parallel algorithms are implemented for coupled PDEs on interface-separated domains. The newly designed scheme is used to solve large-scale 3-D calcium models on neurons.
6:00pm-6:20pm (Oct 1)
CLB 112
Seulip Lee,
University of Georgia

Enriched Galerkin methods for the Stokes equations with modified weak Galerkin bilinear forms
  We propose enriched Galerkin (EG) methods for the Stokes equations using modified weak Galerkin (mWG) bilinear forms. The discrete inf-sup stability is a fundamental property of numerical methods for the Stokes equations, and it has been successfully achieved by recent work on new EG methods whose numerical velocity is a discontinuous piecewise linear function. In the EG methods, interior penalty discontinuous Galerkin (IPDG) approaches have been adopted to weakly impose the continuity of discontinuous numerical velocity, so it is necessary to consider a sufficiently large penalty parameter. For the same numerical velocity, we apply mWG bilinear forms to handle the discontinuity of velocity without such penalty parameter, so our EG method is a parameter-free EG method for the Stokes equations. We prove that our parameter-free EG method guarantees the optimal convergence rates, and pressure-robustness is also achieved by employing a velocity reconstruction operator on the load vector of the right-hand side. Furthermore, the theoretical results are verified by various numerical examples.
10:30am-10:50am (Oct 2)
CLB 112
Yi Zhang,
University of North Carolina at Greensboro
Dual-wind discontinuous Galerkin methods for an elliptic optimal control problem
  Optimal control problems appear in many applications and have received significant attention in recent years. In this talk we study a symmetric dual-wind discontinuous Galerkin method for solving an elliptic optimal control problem with distributed control constraints. We will discuss the motivation of the proposed method and its convergence analysis. Numerical results will be provided to demonstrate the effectiveness of the proposed method.
10:50am-11:10am (Oct 2)
CLB 112
Donsub Rim,
Washington University in St. Louis

Reduced Deep Networks (RDNs) for model reduction of nonlinear waves
  Model reduction of partial differential equations rely on low-dimensional representations of solution manifolds. However, linear dimensionality reduction methods such as proper orthogonal decomposition become inefficient for solution manifolds of wave problems. In this talk, we discuss how deep neural networks are suitable for yielding low-dimensional representations of the solution manifold. We will also discuss how these neural network representations satisfy the key requirements in the classical model reduction framework necessary for obtaining online-offline decomposition. Numerical examples involving the 1D Euler’s equation will be presented.
(This is a joint work with Randall J. LeVeque and Gerrit Welper)
11:10am-11:30am (Oct 2)
CLB 112
Natasha Sharma,
University of Texas at El Paso
A numerical scheme for a two-field model for binary systems containing surfactants

We present a numerical approximation for the binary fluid-surfactant two-field model. This model is mathematically represented by two nonlinearly coupled Cahn-Hilliard equations involving two fields namely, the scalar order parameter representing the difference in the local concentration of the binary fluids and the other field representing the local surfactant concentration. This model proposed by Laradji et al. describes not only the microemusification processes but also captures the effects of surfactants on the phase transition dynamics. Hence in this sense presents an advantage over the sixth-order phase field model capturing the microemulsification processes proposed by Gompper and co-authors. Numerical results will be presented to illustrate the performance of our scheme.

The author acknowledges the financial support of NSF DMS-2110774 and the author thanks Texas Advanced Computing Center for providing the high performance computing resources for the numerical simulations.

11:30am-11:50am (Oct 2)
CLB 112
Yangwen Zhang,
Carnegie Mellon University

A new reduced order model of linear parabolic PDEs
  How to build an accurate reduced order model (ROM) for multidimensional time dependent partial differ- ential equations (PDEs) is quite open. In this paper, we propose a new ROM for linear parabolic PDEs. We prove that our new method can be orders of magnitude faster than standard solvers, and is also much less memory intensive. Under some assumptions on the problem data, we prove that the convergence rates of the new method is the same with standard solvers. Numerical experiments are presented to confirm our theoretical result.