Spatial Mesh Research Articles

Generalized SAV-Exponential Integrator Schemes for Allen--Cahn Type Gradient Flows

The energy dissipation law and the maximum bound principle (MBP) are two important physical features of the well-known Allen-Cahn equation. While some commonly-used first-order time stepping schemes have turned out to preserve unconditionally both energy dissipation law and MBP for the equation, restrictions on the time step size are still needed for existing second-order or even higher-order schemes in order to have such simultaneous preservation. In this paper, we develop and analyze novel first- and second-order linear numerical schemes for a class of Allen-Cahn type gradient flows. Our schemes combine the generalized scalar auxiliary variable (SAV) approach and the exponential time integrator with a stabilization term, while the standard central difference stencil is used for discretization of the spatial differential operator. We not only prove their unconditional preservation of the energy dissipation law and the MBP in the discrete setting, but also derive their optimal temporal error estimates under fixed spatial mesh. Numerical experiments are also carried out to demonstrate the properties and performance of the proposed schemes.

High-order structure-preserving Du Fort–Frankel schemes and their analyses for the nonlinear Schrödinger equation with wave operator

Du Fort–Frankel-type finite difference methods (DFFT-FDMs) are famous for good stability and easy implementation. In this study, by a perfect combination of the classical fourth-order difference to approximate the second-order spatial derivatives with the idea of DFFT-FDMs, a class of high-order structure-preserving DFFT-FDMs (SP-DFFT-FDMs) are firstly developed for solving the periodic initial–boundary value problems (PIBVPs) of one-dimensional (1D) and two-dimensional (2D) nonlinear Schrödinger equations with wave operator (NLSW), respectively. By using the discrete energy method, it is shown that their solutions satisfy the discrete energy- and mass-conservative laws, and are conditionally convergent with an order of O(τ2+hx4+(τhx)2) and O(τ2+hx4+hy4+(τhx)2+(τhy)2) in the discrete H1-norm, respectively. Here, τ denotes time step size, while, hx and hy represent spatial mesh sizes in x and y directions, respectively. Then, by supplementing a stabilized term, a type of stabilized SP-DFFT-FDMs are devised. They not only preserve the discrete energy- and mass-conservation laws, but also own much better stability than original SP-DFFT-FDMs. Finally, numerical results confirm the theoretical findings, and the efficiency of our algorithms.

A conforming sliding mesh technique for an embedded‐hybridized discontinuous Galerkin discretization for fluid‐rigid body interaction

AbstractIn previous work, we introduced a space‐time embedded‐hybridizable discontinuous Galerkin method for the solution of the incompressible Navier–Stokes equations on time‐dependent domains of which the motion of the domain is prescribed. This discretization is exactly mass conserving, locally momentum conserving, and energy‐stable. In this article, we extend this discretization to fluid‐rigid body interaction problems in which the motion of the fluid domain is not known a priori. To account for large rotational motion of the rigid body, we present a novel conforming space‐time sliding mesh technique. For this we introduce a local edge swapping algorithm such that the global mesh of a space‐time slab is one of only four different configurations. These configurations can be pre‐computed thereby reducing any costs associated with changing mesh connectivity. Furthermore, edge swapping occurs only within the space‐time slab between the discrete time‐levels; there is no edge swapping on spatial meshes and so there is no need to project the solution from one spatial mesh to another. We demonstrate the performance of the discretization on various numerical examples.

A metric tensor approach to data assimilation with adaptive moving meshes

Adaptive moving spatial meshes are useful for solving physical models given by time-dependent partial differential equations. However, special consideration must be given when combining adaptive meshing procedures with ensemble-based data assimilation (DA) techniques. In particular, we focus on the case where each ensemble member evolves independently upon its own mesh and is interpolated to a common mesh for the DA update. This paper outlines a framework to develop time-dependent reference meshes using locations of observations and the metric tensors (MTs) or monitor functions that define the spatial meshes of the ensemble members. We develop a time-dependent spatial localization scheme based on the metric tensor (MT localization). We also explore how adaptive moving mesh techniques can control and inform the placement of mesh points to concentrate near the location of observations, reducing the error of observation interpolation. This is especially beneficial when we have observations in locations that would otherwise have a sparse spatial discretization. We illustrate the utility of our results using discontinuous Galerkin (DG) approximations of 1D and 2D inviscid Burgers equations. The numerical results show that the MT localization scheme compares favorably with standard Gaspari-Cohn localization techniques. In problems where the observations are sparse, the choice of common mesh has a direct impact on DA performance. The numerical results also demonstrate the advantage of DG-based interpolation over linear interpolation for the 2D inviscid Burgers equation.

Digital Simulations for Three-dimensional Nonlinear Advection-diffusion Equations Using Quasi-variable Meshes High-resolution Implicit Compact Scheme

A two-level implicit compact formulation with quasi-variable meshes is reported for solving three-dimensions second-order nonlinear parabolic partial differential equations. The new nineteen-point compact scheme exhibit fourth and second-order accuracy in space and time on a variable mesh steps and uniformly spaced mesh points. We have also developed an operator-splitting technique to implement the alternating direction implicit (ADI) scheme for computing the 3D advection-diffusion equation. Thomas algorithm computes each tri-diagonal matrix that arises from ADI steps in minimal computing time. The operator-splitting form is unconditionally stable. The improved accuracy is achieved at a lower cost of computation and storage because the spatial mesh parameters tune the mesh location according to solution values' behavior. The new method is successfully applied to the Navier-Stokes equation, advection-diffusion equation, and Burger's equation for the computational illustrations that corroborate the order, accuracies, and robustness of the new high-order implicit compact scheme. The main highlight of the present work lies in obtaining a fourth-order scheme on a quasi-variable mesh network, and its superiority over the comparable uniform meshes high-order compact scheme.

Digital Simulations for Three-dimensional Nonlinear Advection-diffusion Equations Using Quasi-variable Meshes High-resolution Implicit Compact Scheme

A two-level implicit compact formulation with quasi-variable meshes is reported for solving three-dimensions second-order nonlinear parabolic partial differential equations. The new nineteen-point compact scheme exhibit fourth and second-order accuracy in space and time on a variable mesh steps and uniformly spaced mesh points. We have also developed an operator-splitting technique to implement the alternating direction implicit (ADI) scheme for computing the 3D advection-diffusion equation. Thomas algorithm computes each tri-diagonal matrix that arises from ADI steps in minimal computing time. The operator-splitting form is unconditionally stable. The improved accuracy is achieved at a lower cost of computation and storage because the spatial mesh parameters tune the mesh location according to solution values' behavior. The new method is successfully applied to the Navier-Stokes equation, advection-diffusion equation, and Burger's equation for the computational illustrations that corroborate the order, accuracies, and robustness of the new high-order implicit compact scheme. The main highlight of the present work lies in obtaining a fourth-order scheme on a quasi-variable mesh network, and its superiority over the comparable uniform meshes high-order compact scheme.

Validation of deterministic code OpenNTP for the analysis of the C5G7-1D benchmark with different configurations

This study was conducted with the aim of verifying that the OpenNTP (Open Neutron Transport Package) code is capable of reproducing high-performance and exact calculation results. Therefore, a thorough analysis of the One-dimensional C5G7 benchmark with different configuration of control rods was performed. This benchmark was chosen because of its strong heterogeneity. For the modelling of the pin cell geometry, different sets of spatial meshes are used to study the sensitivity of the multiplication factors on the spatial and angular discretization. In addition, the influence of the control rods has been studied by analysing the scalar fluxes, the power in each pin cell and the power of the assemblies. It was found that assembly power distribution errors were significant for the Unrodded case of the C5G7 benchmark. The calculation results in the present paper have a good agreement with the reference values, which demonstrates that the OpenNTP code can solve neutronics problems accurately.

Optimal Convergence of the Newton Iterative Crank–Nicolson Finite Element Method for the Nonlinear Schrödinger Equation

Abstract An error estimate is presented for the Newton iterative Crank–Nicolson finite element method for the nonlinear Schrödinger equation, fully discretized by quadrature, without restriction on the grid ratio between temporal step size and spatial mesh size. It is shown that the Newton iterative solution converges double exponentially with respect to the number of iterations to the solution of the implicit Crank–Nicolson method uniformly for all time levels, with optimal convergence in both space and time.

Identification of Requirements for FE Modeling of an Adaptive Joining Technology Employing Friction-Spun Joint Connectors (FSJC)

The adaptive joining process employing friction-spun joint connectors (FSJC) is a promising method for the realization of adaptable joints and thus for lightweight construction. In addition to experimental investigations, numerical studies are indispensable tools for its development. Therefore, this paper includes an analysis of boundary conditions for the spatial discretization and mesh modeling techniques, the material modeling, the contact and friction modeling, and the thermal boundary conditions for the finite element (FE) modeling of this joining process. For these investigations, two FE models corresponding to the two process steps were set up and compared with the two related processes of friction stir welding and friction drilling. Regarding the spatial discretization, the Lagrangian approach is not sufficient to represent the deformation that occurs. The Johnson-Cook model is well suited as a material model. The modeling of the contact detection and friction are important research subjects. Coulomb’s law of friction is not adequate to account for the complex friction phenomena of the adaptive joining process. The thermal boundary conditions play a decisive role in heat generation and thus in the material flow of the process. It is advisable to use temperature-dependent parameters and to investigate in detail the influence of radiation in the entire process.

Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems III: Nonlinear Theory

In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE and establish the existence of travelling waves. In particular, we consider the time dependent spatial mesh adaptation method that aims to equidistribute the arclength of the solution under consideration. We assume that this equidistribution is strictly enforced, which leads to the non-local problem with infinite range interactions that we derived in Hupkes and Van Vleck (J Dyn Differ Eqn, 2021). Using the Fredholm theory developed in Hupkes and Van Vleck (J Dyn Differ Eqn, 2021) we setup a fixed point procedure that enables the travelling PDE waves to be lifted to our spatially discrete setting.

High‐order implicit‐explicit additive Runge–Kutta schemes for numerical combustion with adaptive mesh refinement

AbstractThe objective of this study is to develop and apply efficient solution techniques for numerical modeling of combustion with stiff chemical kinetics in practical combustors. The new technique combines a fourth‐order implicit‐explicit (IMEX) additive Runge‐Kutta scheme (ARK) with adaptive mesh refinement (AMR). The IMEX component treats the stiff reactions implicitly but integrates convection and diffusion explicitly in time, and thus permits the solution to advance with larger time‐step sizes than that of explicit time‐marching methods alone. The AMR further adds computational efficiency by effectively placing high spatial resolution meshes in regions with strong gradients, such as flame fronts. The novelty of this study is in the integration of a fourth‐order IMEX ARK method with AMR for a high‐order finite‐volume scheme and the application to solving complex reacting flows governed by the compressible Navier–Stokes equations with very stiff chemistry in a practical combustor geometry. The effectiveness and performance of the adaptive ARK4 is assessed for complex reacting flows by examining properties, such as the presence of shock waves, the time‐scale changes in response to AMR levels, and the size and stiffness of reaction mechanisms for various fuels such as , , and . The new adaptive ARK4 method is verified and validated using a convection‐diffusion‐reaction problem and shock‐driven combustion, respectively. The validated algorithm is then applied to solve the stiff ‐air combustion in a bluff‐body combustor. A significant speedup of three orders of magnitude is achieved in comparison to the standard ERK4 method at the given solution accuracy.

Fast Solution of Fully Implicit Runge--Kutta and Discontinuous Galerkin in Time for Numerical PDEs, Part II: Nonlinearities and DAEs

Fully implicit Runge--Kutta (IRK) methods have many desirable accuracy and stability properties as time integration schemes, but high-order IRK methods are not commonly used in practice with large-scale numerical PDEs because of the difficulty of solving the stage equations. This paper introduces a theoretical and algorithmic framework for solving the nonlinear equations that arise from IRK methods (and discontinuous Galerkin discretizations in time) applied to nonlinear numerical PDEs, including PDEs with algebraic constraints. Several new linearizations of the nonlinear IRK equations are developed, offering faster and more robust convergence than the often-considered simplified Newton, as well as an effective preconditioner for the true Jacobian if exact Newton iterations are desired. Inverting these linearizations requires solving a set of block 2 x 2 systems. Under quite general assumptions, it is proven that the preconditioned 2 x 2 operator's condition number is bounded by a small constant close to one, independent of the spatial discretization, spatial mesh, and time step, and with only weak dependence on the number of stages or integration accuracy. Moreover, the new method is built using the same preconditioners needed for backward Euler-type time stepping schemes, so can be readily added to existing codes. The new methods are applied to several challenging fluid flow problems, including the compressible Euler and Navier--Stokes equations, and the vorticity-streamfunction formulation of the incompressible Euler and Navier--Stokes equations. Up to 10th-order accuracy is demonstrated using Gauss IRK, while in all cases fourth-order Gauss IRK requires roughly half the number of preconditioner applications as required by standard Singly diagonally implicit Runge--Kutta methods.

Modeling calcium dynamics in neurons with endoplasmic reticulum: Existence, uniqueness and an implicit–explicit finite element scheme

Like many other biological processes, calcium dynamics in neurons containing an endoplasmic reticulum is 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. Using the calcium model as an example of this class of ODE-flux boundary interface problems, we prove the existence, uniqueness and boundedness of the solution by applying comparison theorem, fundamental solution of the parabolic operator and a strategy used in Picard’s existence theorem. Then we propose and analyze an efficient implicit–explicit finite element scheme which is implicit for the parabolic operator and explicit for the nonlinear terms. We show that the stability does not depend on the spatial mesh size. Also the optimal convergence rate in H1 norm is obtained. Numerical experiments illustrate the theoretical results.

Canonical states in relativistic continuum theory with the Green's function method: Neutrons in continuum of zirconium giant-halo nuclei

The canonical states in the relativistic continuum Hartree-Bogoliubov theory with Green's function method are obtained by diagonalizing the density matrix on a spatial mesh. Taking the giant halo nucleus $^{128}\mathrm{Zr}$ as an example, the obtained canonical single-particle energies and wave functions are compared in detail with the corresponding calculations with the box-discretized method. The occupation number ${v}_{i}^{2}$ in the canonical basis and the neutrons in continuum ${N}_{c}$ are investigated for the Zr giant halo nuclei. The dependencies on the pairing strength ${V}_{0}$ and on the different density functionals PK1 and NL-SH are also studied. It is found that the calculations with Green's function and box-discretized methods are almost consistent in describing nuclear global properties and the neutrons in continuum, meanwhile the ${N}_{c}$ of giant halo nuclei may heavily depend on the adopted pairing strength as well as the density functional.

Stable rotational symmetric schemes for nonlinear reaction-diffusion equations

In the simulation of biological pattern forming, it has been observed that the numerical solution is more sensitive to the spatial mesh resolution than the temporal one. Such a higher sensitivity to the spatial resolution is mainly originated from an inaccurate approximation of diffusion differential operators, which might violate the rotational symmetry to be seriously erroneous in low spatial resolutions. Also, it has been known that the second-order Crank-Nicolson time-stepping procedure may introduce spurious oscillations when the initial data or the source term is nonsmooth and the temporal step size is set relatively large. This article studies 9-point finite difference schemes for the diffusion operator to enhance the rotational symmetry, employs the variable-θ method to achieve a nonoscillatory second-order time-stepping procedure, and adopts an effective relaxation linear solver to solve the algebraic systems efficiently. The variable-θ method is proved to satisfy the maximum principle, which guarantees that the time-stepping procedure is unconditionally stable. When the successive over-relaxation method with an optimal relaxation parameter is adopted for the algebraic solver, the iteration converges in 2-4 iterations in most time steps. The overall algorithm is second-order in accuracy and scalable in efficiency. Various examples are given to show the accuracy and efficiency of the proposed algorithm for the numerical solution of the system of nonlinear reaction-diffusion equations.

On the Robustness and Efficiency of the Plane-Wave-Enriched FEM with Variable q-Approach on the 2D Room Acoustics Problem

Partition of unity finite element method with plane wave enrichment (PW-FEM) uses a shape function with a set of plane waves propagating in various directions. For room acoustic simulations in a frequency domain, PW-FEM can be an efficient wave-based prediction method, but its practical applications and especially its robustness must be studied further. This study elucidates PW-FEM robustness via 2D real-scale office room problems including rib-type acoustic diffusers. We also demonstrate PW-FEM performance using a sparse direct solver and a high-order Gauss–Legendre rule with a recently developed rule for ascertaining the number of integration points against the classical linear and quadratic FEMs. Numerical experiments investigating mesh size and room geometrical complexity effects on the robustness of PW-FEM demonstrated that PW-FEM becomes more robust at wide bands when using a mesh in which the maximum element size maintains a comparable value to the wavelength of the upper-limit frequency. Moreover, PW-FEM becomes unstable with lower spatial resolution mesh, especially for rooms with complex shape. Comparisons of accuracies and computational costs of linear and quadratic FEM revealed that PW-FEM requires twice the computational time of the quadratic FEM with a mesh having spatial resolution of six elements per wavelength, but it is highly accurate at wide bands with lower memory and with markedly fewer degrees of freedom. As an additional benefit of PW-FEM, the impulse response waveform of quadratic FEM in a time domain was found to deteriorate over time, but the PW-FEM waveform can maintain accurate waveforms over a long time.

High order unconditionally energy stable RKDG schemes for the Swift–Hohenberg equation

We propose unconditionally energy stable Runge–Kutta (RK) discontinuous Galerkin (DG) schemes for solving a class of fourth order gradient flows including the Swift–Hohenberg equation. Our algorithm is geared toward arbitrarily high order approximations in both space and time, while energy dissipation remains preserved for arbitrary time steps and spatial meshes. The method integrates a penalty free DG method for spatial discretization with a multi-stage algebraically stable RK method for temporal discretization by the energy quadratiztion (EQ) strategy. The resulting fully discrete DG method is proven to be unconditionally energy stable. By numerical tests on several benchmark problems we demonstrate the high order accuracy, energy stability, and simplicity of the proposed algorithm.

Virtual LiDAR Simulation as a High Performance Computing Challenge: Toward HPC HELIOS++

The software HELIOS++ simulates the laser scanning of a given virtual scene that can be composed of different spatial primitives and 3D meshes with distinct granularity. The high computational cost of this type of simulation software demands efficient computational solutions. Classical solutions based on GPU are not well suited when irregular geometries compose the scene combining different primitives and physics models because they lead to different computation branches. In this paper, we explore the usage of parallelization strategies based on static and dynamic workload balancing and heuristic optimization strategies to speed up the ray tracing process based on a k-dimensional tree (KDT). Using HELIOS++ as our case study, we analyze the performance of our algorithms on different parallel computers, including the CESGA FinisTerrae-II supercomputer. There is a significant performance boost in all cases, with the decrease in computation time ranging from 89.5% to 99.4%. Our results show that the proposed algorithms can boost the performance of any software that relies heavily on a KDT or a similar data structure, as well as those that spend most of the time computing with only a few synchronization barriers. Hence, the algorithms presented in this paper improve performance, whether computed on personal computers or supercomputers.

“Wall Hack AR”: AR See-through system with LiDAR and VPS

We propose a method to realize the experience of seeing through an occluding wall by using a 3D scanner and an AR (Augmented Reality) device. This system uses the LiDAR (Light Detection And Ranging) scanner of the Apple iPad Pro as a 3D scanner, sending scanned spatial mesh data to the AR device of the experiencer in real-time. By using a VPS (Visual Positioning System) to also share the actual spatial position of the mesh, a receiving AR device can also draw the scan data registered at the correct position in the actual space. Placing the LiDAR scanner behind an occlusion allows an experience as if the user were seeing through a wall.

Development of a gyrokinetic hyperbolic solver based on discontinuous Galerkin method in tokamak geometry

A hyperbolic solver is developed for the gyrokinetic equation in tokamak geometry. Aiming a whole device modeling of fusion plasma surrounded by first walls of tokamak device, an unstructured spatial mesh is introduced. The discontinuous Galerkin (DG) method is used to discretize the gyrokinetic equation on the mesh and test various numerical elements for the discretization. Based on the conservations of physical quantities such as mass, kinetic energy, and toroidal canonical angular momentum in an axisymmetric configuration of toroidal plasma, we investigate the effects of basis functions for the DG method on the numerical solutions. With proper choices of the basis functions and spatial grid resolutions, the conservations of the key physical quantities are shown to be satisfied nearly to machine accuracies in the simplified circular magnetic geometry. Even in realistic tokamak geometry with machine wall boundaries, it is shown that a good conservation property can be demonstrated if the flux across boundaries of a test domain is carefully accounted for. Also, the invariance of the canonical Maxwellian distribution function in time is well satisfied with the developed solver. The effect of weighting functions for the basis is investigated too. Overall, the Maxwellian weighted basis shows a similar conservation property with the polynomial basis. On the other hand, the Maxwellian weighted basis shows better performance in resolving small scale structures in velocity space, which can be utilized to set up an efficient basis set to simulate fine structures with less computational costs. The parallelization of the newly developed solver is also reported. Employing MPI for the parallelization, the solver shows good performances up to a few thousand CPU cores.