Hybrid Finite Difference Research Articles

Hybrid Finite-Difference Physics-Informed Neural Networks Partial Differential Equation Solver for Complex Geometries

The physics-informed neural network (PINN) is effective in solving the partial differential equation (PDE) by capturing the physics constraints as a part of the training loss function through the automatic differentiation (AD). This study proposes the hybrid finite difference with PINN (HFD-PINN) to fully use the domain knowledge. The main idea is to use the finite-difference method (FDM) locally instead of AD in the framework of PINN. We use AD at complex boundaries and FDM in other domains. To avoid the background mesh, we propose HFD-PINN-sdf, which uses the signed distance function (sdf) to avoid the difference scheme from crossing the domain boundary. In this paper, we demonstrate the performance and compare the results with different numbers of collocation points and architectures for the Poisson equation and Burgers equation. We also chose several different finite-difference schemes, including the compact finite-difference and Crank–Nicolson methods, to verify the robustness of HFD-PINN. We take the heat conduction problem and the heat transfer problem on the irregular domain as examples to demonstrate the efficacy. In summary, HFD-PINN is more instructive and efficient when solving PDEs in complex geometries.

Simultaneous space–time Hermite wavelet method for time-fractional nonlinear weakly singular integro-partial differential equations

An innovative simultaneous space–time Hermite wavelet method has been developed to solve weakly singular fractional-order nonlinear integro-partial differential equations in one and two dimensions with a focus whose solutions are intermittent in both space and time. The proposed method is based on multi-dimensional Hermite wavelets and the quasilinearization technique. The simultaneous space–time approach does not fully exploit for time-fractional nonlinear weakly singular integro-partial differential equations. Subsequently, the convergence analysis is challenging when the solution depends on the entire time domain (including past and future time), and the governing equation is combined with Volterra and Fredholm integral operators. Considering these challenges, we use the quasilinearization technique to handle the nonlinearity of the problem and reconstruct it to a linear integro-partial differential equation with second-order accuracy. Then, we apply multi-dimensional Hermite wavelets as attractive candidates on the resulting linearized problems to effectively resolve the initial weak singularity at t=0. In addition, the collocation method is used to determine the tensor-based wavelet coefficients within the decomposition domain. We elaborate on constructing the proposed simultaneous space–time Hermite wavelet method and design comprehensive algorithms for their implementation. Specifically, we emphasize the convergence analysis in the framework of the L2 norm and indicate high accuracy dependent on the regularity of the solution. The stability of the proposed wavelet-based numerical approximation is also discussed in the context of fractional-order nonlinear integro-partial differential equations involving both Volterra and Fredholm operators with weakly singular kernels. The proposed method is compared with existing methods available in the literature. Specifically, we highlighted its high accuracy and compared it with a recently developed hybrid numerical approach and finite difference methods. The efficiency and accuracy of the proposed method are demonstrated by solving several highly intermittent time-fractional nonlinear weakly singular integro-partial differential equations.

On-load magnetic field calculation for linear permanent-magnet actuators using hybrid 2-D finite-element method and Maxwell–Fourier analysis

PurposeThe purpose of this study is to present a novel extended hybrid analytical method (HAM) that leverages a two-dimensional (2-D) coupling between the semi-analytical Maxwell–Fourier analysis and the finite element method (FEM) in Cartesian coordinates.Design/methodology/approachThe proposed model is applied to flat permanent-magnet linear electrical machines with rotor-dual. The magnetic field solution across the entire machine is established by coupling an exact analytical model (AM), designed for regions with relative magnetic permeability equal to unity, with a FEM in ferromagnetic regions. The coupling between AM and FEM occurs bidirectionally (x, y) along the edges separating teeth regions and their adjacent regions through applied boundary conditions.FindingsThe developed HAM yields accurate results concerning the magnetic flux density distribution, cogging force and induced voltage under various operating conditions, including magnetic or geometric parameters. A comparison with hybrid finite-difference and hybrid reluctance network methods demonstrates very satisfactory agreement with 2-D FEM.Originality/valueThe original contribution of this paper lies in establishing a direct coupling between the semi-analytical Maxwell–Fourier analysis and the FEM, particularly at the interface between adjacent regions with differing magnetic parameters.

PARAMETER-UNIFORM HYBRID NUMERICAL SCHEME FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION PROBLEMS WITH DISCONTINUOUS INITIAL CONDITIONS

This article presents a parameter-uniform hybrid numerical technique for singularly perturbed parabolic convection-diffusion problems (SPPCDP) with discontinuous initial conditions (DIC). It utilizes the classical backward-Euler technique for time discretization and a hybrid finite difference scheme ( which is a proper combination of the midpoint upwind scheme in the outer regions and the classical central difference scheme in the interior layer regions(generated by the DIC)) for spatial discretization. The scheme produces parameter-uniform numerical approximations on a piecewise- uniform Shishkin mesh. When the perturbation parameter ε(0 < ε ≪ 1) is small, it becomes difficult to solve these problems using the classical numerical methods (standard central difference or a standard upwind scheme) on uniform meshes because discontinuous initial conditions frequently appear in the solutions of this class of problems. The method is shown to converge uniformly in the discrete supremum with nearly second-order spatial accuracy. The suggested method is subjected to a stability study, and parameter-uniform error estimates are generated. In order to support the theoretical findings, numerical results are presented.

Line-Based High-Order Hybrid Scheme on Unstructured Grids for Compressible Flows

This study explores the use of line-based finite difference techniques on unstructured grids employing high-order stencils for practical aerodynamic applications. The primary focus is on finite-difference explicit and compact spatial discretization up to sixth order. Problems studied include compressible flows containing discontinuities, which necessitated the use of a hybrid finite difference–finite volume approach, where the former is shown to be effective and accurate for smooth regions, while the latter is robust in capturing discontinuities and shocks. To differentiate between discontinuous and smooth zones, two distinct shock indicators are used, one based on pressure gradients and the other on the divergence of velocity. To achieve faster convergence, spatial discretization is applied with line-based alternating direction implicit (ADI) time integration. A key focus of this work is to demonstrate the proposed methodology in flow scenarios relevant to aircraft applications that also test various aspects of the methodology. Consequently, the cases studied are isentropic vortex convection, standing normal shock, transonic NACA0012 airfoil, supersonic cylinder, unsteady flow past tandem circular cylinders, and supersonic flow past a forward-facing step. Comparisons were also done against an existing line-based finite-volume technique for unstructured grids that employed Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) and fifth-order Weighted Essentially NonOscillatory (WENO5) spatial reconstruction schemes. It is observed that the hybrid finite difference–finite volume approach performed well with excellent agreement against available results, sometimes with far fewer degrees of freedom. Furthermore, the implicit ADI scheme is effective over unstructured grids and provides a three-time speed-up over the traditional Runge–Kutta fourth-order scheme.

Numerical approximation of parabolic singularly perturbed problems with large spatial delay and turning point

PurposeSingular perturbation turning point problems (SP-TPPs) involving parabolic convection–diffusion Partial Differential Equations (PDEs) with large spatial delay are studied in this paper. These type of equations are important in various fields of mathematics and sciences such as computational neuroscience and require specialized techniques for their numerical analysis.Design/methodology/approachWe design a numerical method comprising a hybrid finite difference scheme on a layer-adapted mesh for the spatial discretization and an implicit-Euler scheme on a uniform mesh in the temporal variable. A combination of the central difference scheme and the simple upwind scheme is used as the hybrid scheme.FindingsConsistency, stability and convergence are investigated for the proposed scheme. It is established that the present approach has parameter-uniform convergence of OΔτ+K−2(lnK)2, where Δτ and K denote the step size in the time direction and number of mesh-intervals in the space direction.Originality/valueParabolic SP-TPPs exhibiting twin boundary layers with large spatial delay have not been studied earlier in the literature. The presence of delay portrays an interior layer in the considered problem’s solution in addition to twin boundary layers. Numerical illustrations are provided to demonstrate the theoretical estimates.

A hybrid discrete exterior calculus and finite difference method for anelastic convection in spherical shells

The present work develops, verifies, and benchmarks a hybrid discrete exterior calculus and finite difference (DEC-FD) method for density-stratified thermal convection in spherical shells. Discrete exterior calculus (DEC) is notable for its coordinate independence and structure preservation properties. The hybrid DEC-FD method for Boussinesq convection has been developed by Mantravadi et al. (2023). Motivated by astrophysics problems, we extend this method assuming anelastic convection, which retains density stratification; this has been widely used for decades to understand thermal convection in stars and giant planets. In the present work, the governing equations are splitted into surface and radial components and discrete anelastic equations are derived by replacing spherical surface operators with DEC and radial operators with FD operators. The novel feature of this work is the discretization of anelastic equations with the DEC-FD method and the assessment of a hybrid solver for density-stratified thermal convection in spherical shells. The discretized anelastic equations are verified using the method of manufactured solution (MMS). We performed a series of three-dimensional convection simulations in a spherical shell geometry and examined the effect of density ratio on convective flow structures and energy dynamics. The present observations are in agreement with the benchmark models.

A hybrid finite difference level set–implicit mesh discontinuous Galerkin method for multi-layer coating flows

A mathematical model and numerical framework are presented for computing multi-physics multi-layer coating flow dynamics, with applications to the leveling of multi-layer paint films. The algorithm combines finite difference level set methods and high-order accurate sharp-interface implicit mesh discontinuous Galerkin methods to capture a complex set of multi-physics, incorporating Marangoni-driven multi-phase interfacial flow and the transport, mixing, and evaporation of multiple dissolved species. In particular, we develop several numerical methods for this multi-physics problem, including: high-order local discontinuous Galerkin methods for Poisson problems with Robin boundary conditions on implicitly-defined domains, to capture solvent evaporation; finite difference surface gradient methods, to robustly and accurately incorporate Marangoni stresses; and a coupled multi-physics time stepping approach, to incorporate all the different solvers at play including quasi-Newtonian fluid flow. The framework is applicable to an arbitrary number of layers and dissolved species; here, we apply it in a variety of settings, including multi-solvent evaporative paint dynamics, the flow and leveling of multi-layer automobile paint coatings in both 2D and 3D, and an examination of interfacial turbulence within a multi-layer matter cascade. Our results reproduce several phenomena observed in experiment, such as the formation of Marangoni plumes and Bénard cells. We also use the model to study the impact of long-wave deformational surface modes on immersed interfaces as well as the emergence of the final multi-layer film profile.

Inversion for 3D Conductivity and Chargeability Models Using EM Data Acquired by the New Airborne TargetEM System in Ontario, Canada

This paper introduces an original approach to the joint inversion of airborne electromagnetic (EM) data for three-dimensional (3D) conductivity and chargeability models using hybrid finite difference (FD) and integral equation (IE) methods. The inversion produces a 3D model of physical parameters, which includes conductivity, chargeability, time constant, and relaxation coefficients. We present the underlying principles of this approach and an example of a high-resolution inversion of the data acquired by a new active time domain airborne EM system, TargetEM, in Ontario, Canada. The new TargetEM system collects high-quality multicomponent data with low noise, high power, and a small transmitter–receiver offset. This airborne system and the developed advanced inversion methodology represent a new effective method for mineral resource exploration.

Efficient least-squares reverse time migration in TTI media using a finite-difference solvable pure qP-wave equation

Abstract The anisotropic characteristics of underground media play a crucial role in seismic wave propagation and should be accounted for during seismic imaging. The least-squares reverse time migration (LSRTM) method achieves ideal imaging results by suppressing migration artifacts and balancing amplitude. Therefore, pure quasi-P (P-qP) wave equations in tilted transversely isotropic (TTI) media have been widely used to implement LSRTM, owing to their ability to produce stable and noise-free wavefields. However, solving the anisotropic P-qP-wave equations typically necessitates the use of spectral-based methods, making them computationally inefficient, especially in 3D applications. In this study, we first develop a P-qP-wave equation in TTI media that can be efficiently computed through the finite-difference (FD) approach. Numerical tests show that, in comparison to the previous TTI P-qP-wave equation, the newly derived FD solvable TTI P-qP-wave equation yields reasonably accurate and highly efficient wavefield simulations. Then, building on our newly derived wave equation, we derive its adjoint migration and demigration operator to implement TTI LSRTM. Two synthetic examples suggest that the newly presented P-qP-wave TTI LSRTM approach effectively correct for the anisotropy effects, providing high-quality imaging results. Additionally, our approach has superior computational efficiency over the conventional P-qP-wave TTI LSRTM technique based on a hybrid FD pseudo-spectral (HFDPS) solver.

Sixth-order hybrid finite difference methods for elliptic interface problems with mixed boundary conditions

In this paper, we develop sixth-order hybrid finite difference methods (FDMs) for the elliptic interface problem −∇⋅(a∇u)=f in Ω﹨Γ, where Γ is a smooth interface inside Ω. The variable scalar coefficient a>0 and source f are possibly discontinuous across Γ. The hybrid FDMs utilize a 9-point compact stencil at any interior regular points of the grid and a 13-point stencil at irregular points near Γ. For interior regular points away from Γ, we obtain a sixth-order 9-point compact FDM satisfying the sign and sum conditions for ensuring the M-matrix property. We also derive sixth-order compact (4-point for corners and 6-point for edges) FDMs satisfying the sign and sum conditions for the M-matrix property at any boundary point subject to (mixed) Dirichlet/Neumann/Robin boundary conditions. Thus, for the elliptic problem without interface (i.e., Γ is empty), our compact FDM has the M-matrix property for any mesh size h>0 and consequently, satisfies the discrete maximum principle, which guarantees the theoretical sixth-order convergence. For irregular points near Γ, we propose fifth-order 13-point FDMs, whose stencil coefficients can be effectively calculated by recursively solving several small linear systems. Theoretically, the proposed high order FDMs use high order (partial) derivatives of the coefficient a, the source term f, the interface curve Γ, the two jump functions along Γ, and the functions on ∂Ω. Numerically, we always use function values to approximate all required high order (partial) derivatives in our hybrid FDMs without losing accuracy. Our proposed FDMs are independent of the choice representing Γ and are also applicable if the jump conditions on Γ only depend on the geometry (e.g., curvature) of the curve Γ. Our numerical experiments confirm the sixth-order convergence in the l∞ norm of the proposed hybrid FDMs for the elliptic interface problem.

A second order numerical method for singularly perturbed Volterra integro-differential equations with delay

Abstract This study deals with singularly perturbed Volterra integro-differential equations with delay. Based on the properties of the exact solution, a hybrid difference scheme with appropriate quadrature rules on a Shishkin-type mesh is constructed. By using the truncation error estimate techniques and a discrete analogue of Grönwall’s inequality it is proved that the hybrid finite difference scheme is almost second order accurate in the discrete maximum norm. Numerical experiments support these theoretical results and indicate that the estimates are sharp.

Hybrid conservative central/WENO finite difference scheme for two-dimensional detonation problems

In the present work, a hybrid finite difference scheme is proposed and tested for the study of two-dimensional detonation problems. The hybrid scheme consists of a fifth-order weighted essentially nonoscillatory (WENO) method for discontinuous regions and a fourth or sixth-order robust conservative central finite difference scheme for the remaining domain. An arbitrary shock sensor is used to identify the discontinuities. The proposed shock sensor is based on the absolute value of the density gradient with an arbitrary threshold computed at each time step using a fast image segmentation technique. The sensor is complemented with a Ducros sensor to avoid the selection of vortices as discontinuities. The hybrid scheme is tested for several benchmark examples, including a two-dimensional transverse detonation wave, detonation wave diffraction, and a multiple obstacles test case. The obtained results show that the proposed sensor detects discontinuities adequately and the hybrid schemes give the same results as the fifth-order WENO in faster computational time.

A hybrid discrete exterior calculus and finite difference method for Boussinesq convection in spherical shells

We present a new hybrid discrete exterior calculus (DEC) and finite difference (FD) method to simulate fully three-dimensional Boussinesq convection in spherical shells subject to internal heating and basal heating, relevant in the planetary and stellar phenomenon. We employ DEC to compute the surface spherical flows, taking advantage of its unique features including coordinate system independence to preserve the spherical geometry, while we discretize the radial direction using FD method. The grid employed for this novel method is free of problems like the coordinate singularity, grid non-convergence near the poles, and the overlap regions. We have developed a parallel in-house code using the PETSc framework to verify the hybrid DEC-FD formulation and demonstrate convergence. We have performed a series of numerical tests which include quantification of the critical Rayleigh numbers for spherical shells characterized by aspect ratios ranging from 0.2 to 0.8, simulation of robust convective patterns in addition to stationary giant spiral roll covering all the spherical surface in moderately thin shells near the weakly nonlinear regime, and the quantification of Nusselt and Reynolds numbers for basally heated spherical shells.

An almost second-order robust computational technique for singularly perturbed parabolic turning point problem with an interior layer

The subject of this paper is the numerical investigation of a singularly perturbed parabolic problem with a simple interior turning point on a rectangular domain in the x−t plane with Dirichlet boundary conditions. At the location of the turning point, the solution to this class of problems has an internal layer. By decomposing the solution into a regular and singular component, we may first establish stronger bounds for the regular and singular components, as well as their derivatives. The fitted mesh finite difference scheme, which consists of a hybrid finite difference operator on a non-uniform Shishkin mesh in the space direction and a backward implicit Euler scheme on a uniform mesh in the time variable, is used to identify the approximate numerical solution. The accuracy of the resulting scheme is then improved in the temporal direction using the Richardson extrapolation scheme, which reveals that the extrapolated scheme is almost second-order uniformly convergent in both space and time variables when ɛ≤CN−1. Two numerical examples are used to demonstrate the efficacy and accuracy of the finite difference approaches mentioned.

A hybrid FDTD/MoM algorithm with a conformal Huygens' equivalent surface for MRI RF coil design and analysis

Accurate design and analysis of radiofrequency (RF) coils are crucial for ultra-high field (UHF) magnetic resonance imaging (MRI) applications. To improve the numerical accuracy of electromagnetic (EM) simulations, we propose a hybrid finite difference time domain (FDTD)/method of moments (MoM) method. Unlike conventional cuboid-like Huygens' equivalent surfaces (HES), we proposed to use a conformal HES to interface the EM data of the FDTD and MoM zone. The shape and size of the conformal surface can be adjusted to fit different RF coil models, thus broadening the application range of the hybrid FDTD/MoM method. Two numerical models: an 8-channel ellipse array, and an 8-channel bent dipole array, are simulated and compared with the conventional HES counterpart. Numerical results demonstrate the capability of the conformal HES method in the analysis of RF coils.

Theoretical Error Analysis of Hybrid Finite Difference–Asymptotic Interpolation Method for Non-Newtonian Fluid Flow

In this paper, we utilized a hybrid method for the unsteady flow of the non-Newtonian third-grade fluid that combines the finite difference with the asymptotic interpolation method. This hybrid method is used to satisfy the semiunbound domain condition of the fluid flow’s length approaching infinity. The primary issue with this research is how much of the hybrid approach’s error may be accepted to guarantee that the method is significant. This paper discussed theoretical error analysis for numerical solutions, including the range and norm of error. The perturbation method’s concept is used to assess the hybrid method’s error. It is discovered that the hybrid approach’s relative error norm is lower than that of the finite difference method. In terms of the error standard, the hybrid approach is more consistent. Error analysis is performed to check for the accuracy as well as the platform for variable mesh size finite difference method in the future research.

Uniformly convergent hybrid numerical scheme for singularly perturbed turning point problems with delay

In the present work, we consider a class of singularly perturbed turning point problems with delay for their numerical solution. Due to the presence of a delay, there occurs an interior layer in the solution along with twin boundary layers (SIAM J. Appl. Math. 42(3): 502–530). Some a priori estimates are given on the exact solution and its derivatives. A numerical technique composed of hybrid finite difference scheme on a layer-adapted Shishkin mesh is proposed. We establish the consistency, stability and convergence of the proposed technique. To validate the theoretical predictions, we present some numerical illustrations. The proposed scheme is proved to have an almost second-order convergence rate, uniform with respect to the perturbation parameter ε.

A Shock Sensor Based on Image Segmentation with Application to a Hybrid Central/WENO Scheme

A novel shock sensor based on image segmentation is proposed for flows with shock or detonation waves. It consists of a function computed based on the numerical Schliren formulation, which is the absolute value of the density gradient. A fast segmentation technique is applied to the sensor function to determine the threshold of the sensor. The candidate troubled cells detected via the computed threshold are further filtered by a Ducros sensor and multiresolution analysis to exclude turbulent zones from the troubled cells. The proposed sensor is applied to a finite difference hybrid scheme, tested for several cases, and compared with other sensors, namely the Ducros sensor, the multiresolution analysis, and WENO- and TENO-based sensors. The results show that the proposed sensor detects the shock and detonation waves more accurately than the other sensors with fewer cells and reduces the computational time of the hybrid scheme.

An efficient hybrid multi-resolution WCNS scheme for solving compressible flows

A hybrid finite difference fifth-order multi-resolution weighted compact nonlinear scheme (WCNS) is presented for simulating the inviscid and viscous compressible flows in this paper. This scheme borrows and improves the idea of the recently developed multi-resolution WENO scheme for node-to-midpoint nonlinear interpolation procedures of the WCNS scheme, which implements higher-order approximate interpolation at midpoints based on a series of nested stencils. This strategy can achieve the expected design accuracy under different linear weights, and can gradually decrease from high-order accuracy to first-order accuracy near the discontinuities. In addition, a hybridization strategy is proposed to further improve the computational efficiency of the presented multi-resolution WCNS scheme. The strategy does not depend on a specific problem to be solved, nor does it contain any problem-related artificial parameters. Finally, some benchmark examples are used to verify the performance of the presented scheme in terms of numerical accuracy, shock capture capability, robustness, computational efficiency, etc.