Special Session 104: Recent Developments in High-Order Numerical Methods for Multiscale/Multiphysics Partial Differential Equations

Discontinuous Galerkin approximation of the stationary Boussinesq system with a Navier-type boundary condition

Afaf Bouharguane University of Bordeaux France Co-Author(s):    Nour Seloula

Abstract: In this talk, we introduce and analyze a discontinuous Galerkin method for approximating the solutions of the stationary Boussinesq system, which models non-isothermal fluid flow. The model consists of incompressible Navier-Stokes equations, which describe the velocity and pressure of the fluid, coupled with an advection-diffusion equation for the temperature. We impose a Navier-type boundary condition on the velocity and a Dirichlet boundary condition on the temperature. The proposed numerical scheme combines an interior penalty discontinuous Galerkin method with an upwind approach for the Boussinesq system. We prove existence and uniqueness results for the discrete scheme under a certain regularity assumption of the domain. A priori error estimates in terms of natural energy norms for the velocity, pressure, and temperature are also derived. We conclude with some numerical experiments.

Well-balanced positivity-preserving high-order discontinuous Galerkin methods for Euler equations with gravitation

Jie Du East China Normal University Peoples Rep of China Co-Author(s):

Abstract: We develop high order discontinuous Galerkin (DG) methods with Lax-Friedrich fluxes for Euler equations under gravitational fields. Such problems may yield steady-state solutions and the density and pressure are positive. There were plenty of previous works developing either well-balanced (WB) schemes to preserve the steady states or positivity-preserving (PP) schemes to get positive density and pressure. However, it is rather difficult to construct WB PP schemes with Lax-Friedrich fluxes, due to the penalty term in the flux. In fact, for general PP DG methods, the penalty coefficient must be sufficiently large, while the WB scheme requires that to be zero. This contradiction can hardly be fixed following the original design of the PP technique. In this talk, we reformulate the source term such that it balances with the flux term when the steady state has reached. To obtain positive numerical density and pressure, we choose a special penalty coefficient in the Lax-Friedrich flux, which is zero at the steady state. The technique works for general steady-state solutions with zero velocities. Numerical experiments are given to show the performance of the proposed methods.

Compact difference finite element method for high-dimensional convection-diffusion equations

xinlong feng xinjiang university Peoples Rep of China Co-Author(s):

Abstract: In this work, a difference finite element (DFE) method is proposed for solving 3D steady convection-diffusion equations that can maximize good applicability and efficiency of both FDM and FEM. The essence of this method lies in employing the centered difference discretization in the $z$-direction and the FE discretization based on the $P_1$ conforming elements in the $(x,y)$ plane. This allows us to solve PDEs on complex cylindrical domains at lower computational costs compared to applying 3D FEM. We derive the stability estimates for the DFE solution and establish the explicit dependence of $H_1$ error bounds on the diffusivity, convection field modulus, and mesh size. Moreover, a compact DFE method is presented for the similar problems. Finally, we provide numerical examples to verify the theoretical predictions and showcase the accuracy of the considered method.

Multiphysics finite element method for thermo-poroelasticity

Zhihao Ge Henan University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we propose a multiphysics finite element method for a thermo-poroelasticity model. To reveal the multi-physical processes of deformation, diffusion and heat transfer and propose some stable numerical methods, we reformulate the original model into a fluid coupled problem. Then, we propose a fully discrete finite element method based on the multiphysics reformulation and the backward Euler method for time discretization. And we give the stability analysis and convergence order of the above proposed method. Finally, we show some numerical examples to verify the theoretical results.

Solve electromagnetic interface problems on unfitted meshes

Ruchi Guo Sichuan University Peoples Rep of China Co-Author(s):    Long Chen, Shuhao Can

Abstract: Electromagnetic interface problems widely appear in a lot of engineering applications, such as electric actuators, invasive detection techniques and integrated circuit, which are typically described by Maxwell equations with discontinuous coefficients. Conventional finite element methods require a body-fitted mesh to solve interface problems, but generating a high-quality mesh for complex interface geometry is usually very expensive. Instead, using unfitted mesh finite element methods can circumvent mesh generation procedure, which greatly improves the computational efficiency. However, the low regularity of Maxwell equations makes its computation very sensitive to the conformity of the approximation spaces. This very property poses challenges on unfitted mesh finite element methods, as most of them resort to non-conforming spaces. In this talk, we will present our recent progress including several methods for this topic.

Modelling of compressible multi-component two-phase flow with multi-component Navier boundary condition

Qiaolin He Sichuan University Peoples Rep of China Co-Author(s):    Junkai Wang

Abstract: In this work, we derive a dimensionless model for compressible multi-component two-phase flows with Peng-Robinson equation of state (EoS), incorporated with the multi-component Navier boundary condition (MNBC). We propose three linearly decoupled and energy-stable numerical schemes for this model. These schemes are developed based on the Lagrange multiplier approach for bulk Helmholtz free energy and surface free energy. One of them is based on a component-wise approach, which requires solving a sequence of linear, separate mass balance equations and leads to an original discrete energy that unconditionally dissipates. Another numerical scheme is based on a component-separate approach, which requires solving a sequence of linear, separate mass balance equations but leads to a modified discrete energy dissipating under certain conditions. Numerical results are presented to verify the effectiveness of the proposed methods.

An Asymptotic-Preserving and Energy-Conserving Particle-In-Cell Method for Vlasov-Maxwell Equations

Lijie Ji Shanghai University Peoples Rep of China Co-Author(s):    Zhiguo Yang, Zhuoning Li, Dong Wu, Shi Jin, and Zhenli Xu

Abstract: In this talk, we will introduce an asymptotic-preserving and energy-conserving (APEC) Particle-In-Cell (PIC) algorithm for the Vlasov-Maxwell system. This algorithm not only guarantees that the asymptotic limiting of the discrete scheme is a consistent and stable discretization of the quasi-neutral limit of the continuous model, but also preserves Gauss`s law and energy conservation at the same time. Thus, it is promising to provide stable simulations of complex plasma systems even in the quasi-neutral regime. The key ingredients for achieving these properties include the generalized Ohm`s law for the electric field, ensuring that the asymptotic-preserving discretization can be achieved, and a proper decomposition of the effects of the electromagnetic fields, allowing a Lagrange multiplier method to be appropriately employed for correcting the kinetic energy. We investigate the performance of the APEC method with three benchmark tests in one dimension, including the linear Landau damping, the bump on-tail problem, and the two-stream instability. Detailed comparisons are conducted by including the results from the classical explicit leapfrog and the previously developed asymptotic-preserving PIC schemes. The numerical experiments show that the proposed APEC scheme can provide accurate and stable simulations in both kinetic and quasi-neutral regimes, demonstrating the attractive properties of the method across scales.

Stabilized numerical simulations for the transport equation in a fluid

Seulip Lee Tufts University USA Co-Author(s):    Shuhao Cao, Long Chen

Abstract: This talk presents stabilized numerical simulations for the transport equation in a fluid while applying polygonal discretization. A convection-dominated problem explains convective and molecular transport along a given fluid velocity with a small diffusive effect, where classical numerical methods may yield spurious oscillations on numerical solutions and fail to provide accurate simulations. The edge-averaged finite element (EAFE) scheme is a stable discretization for the convection-dominated problem, and its stability is mathematically verified by the discrete maximum principle (DMP). We aim to generalize the edge-averaged stabilization to a polygonal discretization called the virtual element method. Hence, the edge-averaged virtual element (EAVE) methods produce stable and accurate numerical simulations on polygonal meshes and have less computational complexity than other stabilized schemes on polygons. We also show numerical experiments with numerical solutions with sharp boundary layers.

A new type of simplified inverse Lax-Wendroff boundary treatment for hyperbolic conservation law

Shihao Liu KTH Royal Institute of Technology Sweden Co-Author(s):    Tingting Li, Ziqiang Cheng, Yan Jiang, Chiwang Shu, Mengping Zhang

Abstract: We design a new kind of high order inverse Lax-Wendroff (ILW) boundary treatment for solving hyperbolic conservation laws with finite difference method on a Cartesian mesh. This new ILW method decomposes the construction of ghost point values near inflow boundary into two steps: interpolation and extrapolation. First, we impose values of some artificial auxiliary points through a polynomial interpolating the interior points near the boundary. Then, we will construct a Hermite extrapolation based on those auxiliary point values and the spatial derivatives at boundary obtained via the ILW procedure. This polynomial will give us the approximation to the ghost point value. By an appropriate selection of those artificial auxiliary points, high-order accuracy and stable results can be achieved. Moreover, theoretical analysis indicates that comparing with the original ILW method, especially for higher order accuracy, the new proposed one would require fewer terms using the relatively complicated ILW procedure and thus improve computational efficiency on the premise of maintaining accuracy and stability. We perform numerical experiments on several benchmarks, including one- and two-dimensional scalar equations and systems. The robustness and efficiency of the proposed scheme is numerically verified.

Explicit Runge-Kutta methods for quadratic optimization via gradient flow equations

Tuo Liu King Abdullah University of Science and Technology Saudi Arabia Co-Author(s):    Tuo Liu, David Ketcheson

Abstract: This talk focuses on analyzing and developing accelerated optimization methods for the class of smooth and strongly convex functions. We derive a family of optimal gradient-based methods (named OERKD) for quadratic programming by building a mapping between continuous dynamics and discrete updates of optimization algorithms. Optimality of convergent rates is proved by analysis of explicit Runge-Kutta methods on the gradient flow equation with linear stability conditions. Experiments demonstrate the effectiveness of the proposed algorithm even on classical nonlinear problems. A noteworthy byproduct is proving the asymptotic equivalence between OERKD and Polyak`s heavy ball method, which subtly bridges two primary integration schemes.

Efficient and Parallel Solution of High-order Continuous Time Galerkin for Dissipative and Wave Propagation Problems

Yong Liu Academy of Mathematics and Systems Science, CAS Peoples Rep of China Co-Author(s):    Zhiming Chen

Abstract: In this talk, I will propose efficient and parallel algorithms for the implementation of the high-order continuous time Galerkin method for dissipative and wave propagation problems. By using Legendre polynomials as shape functions, we obtain a special structure of the stiffness matrix that allows us to extend the diagonal Pad\`e approximation to solve ordinary differential equations with source terms. The unconditional stability, hp error estimates, and hp superconvergence at the nodes of the continuous time Galerkin method are proved. Numerical examples will be shown to confirm our theoretical results.

A moment-based Hermite WENO scheme with unified stencils for hyperbolic conservation laws

Jianxian QIU Xiamen University Peoples Rep of China Co-Author(s):    Chuan Fan, Zhuang Zhao

Abstract: In this presentation, we introduce a fifth-order moment-based Hermite weighted essentially non-oscillatory scheme with unified stencils (termed as HWENO-U) for hyperbolic conservation laws. The main idea of the HWENO-U scheme is to modify the first-order moment by a HWENO limiter only in the time discretizations using the same information of spatial reconstructions, in which the limiter not only overcomes spurious oscillations well, but also ensures the stability of the fully-discrete scheme. For the HWENO reconstructions, a new scale-invariant nonlinear weight is designed by incorporating only the integral average values of the solution, which keeps all properties of the original one while is more robust for simulating challenging problems with sharp scale variations. Compared with previous HWENO schemes, the advantages of the HWENO-U scheme are: (1) a simpler implemented process involving only a single HWENO reconstruction applied throughout the entire procedures without any modifications for the governing equations; (2) increased efficiency by utilizing the same candidate stencils, reconstructed polynomials, and linear and nonlinear weights in both the HWENO limiter and spatial reconstructions; (3) reduced problem-specific dependencies and improved rationality, as the nonlinear weights are identical for the function $u$ and its non-zero multiple $\zeta u$. Besides, the proposed scheme retains the advantages of previous HWENO schemes, including compact reconstructed stencils and the utilization of artificial linear weights. Extensive benchmarks are carried out to validate the accuracy, efficiency, resolution, and robustness of the proposed scheme.

A local discontinuous Galerkin method for the Novikov equation

Qi Tao Beijing University of Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, we present a local discontinuous Galerkin method for the Novikov equation that contains cubic nonlinear high-order derivatives. Flux correction techniques are used to ensure the stability of the numerical scheme. The H^1-norm stability of the general solution and the error estimate for smooth solutions without using any priori assumptions are presented. Numerical examples demonstrate the accuracy and capability of the LDG method for solving the Novikov equation.

Optimal L2 error estimates of unconditionally stable FE schemes for the Cahn-Hilliard-Navier-Stokes system

Jilu Wang Harbin Institute of Technology (Shenzhen) Peoples Rep of China Co-Author(s):

Abstract: The paper is concerned with the analysis of a popular convex-splitting finite element method for the Cahn-Hilliard-Navier-Stokes system, which has been widely used in practice. Since the method is based on a combined approximation to multiple variables involved in the system, the approximation to one of the variables may seriously affect the accuracy for others. Optimal-order error analysis for such combined approximations is challenging. The previous works failed to present optimal error analysis in $L^2$-norm due to the weakness of the traditional approach. Here we first present an optimal error estimate in $L^2$-norm for the convex-splitting FEMs. We also show that optimal error estimates in the traditional (interpolation) sense may not always hold for all components in the coupled system due to the nature of the pollution/influence from lower-order approximations. Our analysis is based on two newly introduced elliptic quasi-projections and the superconvergence of negative norm estimates for the corresponding projection errors. Numerical examples are also presented to illustrate our theoretical results. More important is that our approach can be extended to many other FEMs and other strongly coupled phase field models to obtain optimal error estimates.

A penalty free weak Galerkin finite element method on quadrilateral meshes

Ruishu Wang Jilin University Peoples Rep of China Co-Author(s):    Jiangguo Liu; Zhuoran Wang

Abstract: The weak Galerkin finite element methods are non-standard finite element methods. The newly defined weak functions are considered as the approximate functions, which have two parts, inner and boundary, on each element. Weak derivatives are correspondingly defined. Appropriate spaces should be used when no penalty term is employed. We use the Arbogast-Correa element to define the weak gradient and obtain a penalty-free weak Galerkin scheme, which is then employed to solve problems related to Stokes flow, linear elasticity, and poroelasticity.

The Finite volume element method with global conservation law

Xiang Wang Jilin University Peoples Rep of China Co-Author(s):    Xinyuan Zhang

Abstract: Conservation laws are fundamental physical properties that are expected to be preserved in numerical discretizations. We propose a two-layered dual strategy for the finite volume element method (FVEM), which possesses the conservation laws in both flux form and equation form. In particular, for problems with Dirichlet boundary conditions, the proposed schemes preserves conservation laws on all triangles, whereas conservation properties may be lost on boundary dual elements by existing vertex-centered finite volume element schemes. Theoretically, we carry out the optimal $L^2$ analysis with reducing the regularity requirement from $u\in H^{k+2}$ to $u\in H^{k+1}$. While, as a comparison, all existing $L^2$ results for high-order $(k>=2)$ finite volume element schemes require $u\in H^{k+2}$ in the analysis.

Recent Advances in High-Order Bound-Preserving Schemes and Theory

Kailiang Wu Southern University of Science and Technology Peoples Rep of China Co-Author(s):

Abstract: Solutions to many partial differential equations (PDEs) are subject to certain bounds or constraints. For instance, in fluid dynamics, density and pressure must remain positive, while in relativistic cases, fluid velocity must not exceed the speed of light. Developing bound-preserving numerical methods that uphold these intrinsic constraints is crucial. Recently, significant attention has been given to design provably bound-preserving schemes, though challenges remain, particularly for systems with nonlinear constraints. In this talk, I will present our recent efforts in developing high-order bound-preserving schemes and theories: 1. Geometric Quasilinearization (GQL): Drawing on key insights from geometry, we propose a novel and general framework called geometric quasilinearization. GQL offers an effective approach for addressing bound-preserving problems with nonlinear constraints by transforming these constraints into linear ones through the introduction of auxiliary variables. We establish the fundamental principles and general theory of GQL using the geometric properties of convex regions and present three effective methods for constructing GQL. 2. Optimal Cell Average Decomposition (OCAD): Utilizing convex geometry and symmetric group theory, we develop the optimal cell average decomposition theory, which provides a foundation for constructing more efficient bound-preserving schemes. We demonstrate that the classic Zhang-Shu CAD is optimal in one dimension but generally not in multiple dimensions, thereby addressing their conjecture proposed in 2010. We apply the GQL and OCAD approaches to various PDEs, showcasing their effectiveness and advantages through diverse and challenging examples and applications, including magnetohydrodynamics (MHD), relativistic hydrodynamics, and the ten-moment Gaussian closure system.