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High order geometric methods with splines: fast solution with explicit time-stepping for Maxwell equations
Bernard Kapidani, Rafael Vázquez
TL;DR
High-order Maxwell discretizations can be computationally costly when each time step requires a mass-matrix solve. This paper combines spline de Rham complexes with discrete Hodge operators to form an explicit-in-time method using Kronecker-structured systems. The authors report substantially lower timestep cost while maintaining accuracy and conservation properties.
Problem
High-order Maxwell methods can be computationally inefficient because they require solving a mass-matrix system at each time step.
Source: §1 · p. 2Method
Results
The pairing-matrix system reduces computational cost per time step by two orders of magnitude compared with solving a mass-matrix system, while maintaining discretization accuracy and conserving charges and energy.
Source: §6 · p. 20Takeaways & Limitations
The spline geometric framework provides a high-order Maxwell method that preserves the system’s geometry and conservation properties while reducing timestep cost.
Source: §6 · p. 20The third local Hodge operator considered in earlier work is discarded because it does not seem to maintain the same approximation properties when mapping from the dual to the primal complex and would not guarantee energy conservation.
Source: §3.2 · p. 7
Abstract
from arXiv · showhide
We introduce a high-order spline geometric approach for the initial boundary value problem for Maxwell's equations. The method is geometric in the sense that it discretizes in structure preserving fashion the two de Rham sequences of differential forms involved in the formulation of the continuous system. Both the Ampere--Maxwell and the Faraday equations are required to hold strongly, while to make the system solvable two discrete Hodge star operators are used. By exploiting the properties of the chosen spline spaces and concepts from exterior calculus, a non-standard explicit in time formulation is introduced, based on the solution of linear systems with matrices presenting Kronecker product structure, rather than mass matrices as in the standard literature. These matrices arise from the application of the exterior (wedge) product in the discrete setting, and they present Kronecker product structure independently of the geometry of the domain or the material parameters. The resulting scheme preserves the desirable energy conservation properties of the known approaches. The computational advantages of the newly proposed scheme are studied both through a complexity analysis and through numerical experiments in three dimensions.
1 Introduction
The introduction motivates geometric spline methods as a route to high-order, structure-preserving Maxwell solvers, then presents an explicit-in-time approach whose timestep systems have Kronecker structure independent of geometry and material parameters.
1 Introduction
- Geometric methods preserve Maxwell conservation properties, but high-order FEM can be computationally inefficient because it requires a mass-matrix solve at each time step.The introduction also notes that some DG flux choices sacrifice energy conservation, while others introduce spurious solutions.Source: §1 · p. 2
- Earlier spline work built dual de Rham complexes and showed high-order convergence for elliptic problems and spurious-free Maxwell eigenvalue results.The present paper builds on that framework to address hyperbolic systems.Source: §1 · p. 2
- The proposed Maxwell method uses Kronecker-structured timestep systems whose structure is independent of domain geometry and material parameters.The systems arise from spline-space wedge products and are solved using LU factorizations of Gram matrices between univariate spline spaces.Source: §1 · p. 2
- The paper develops the spline tools, compares discrete Hodge choices, analyzes efficient explicit time stepping, and validates approximation and efficiency numerically.The sections proceed from mathematical preliminaries to discretization, solver structure, three-dimensional experiments, and concluding remarks.Source: §1 · p. 3
2 Preliminaries
The preliminaries set up Maxwell’s equations in differential forms and define the spline complexes and pairing machinery used in the discretization.
2.1 Maxwell’s equations and differential forms
- For a perfect electrical conductor boundary, electric and magnetic fields and their associated displacement and induction forms are assigned to spaces in the two de Rham sequences.The electric field and magnetic induction use primal spaces with vanishing boundary conditions; the magnetic field and electric displacement use dual spaces.Source: §2.1 · p. 3
2.2 Hodge star operators
- The Hodge star maps k-forms to (3−k)-forms and is defined using a weighted L2 inner product and the exterior product.For the stated bounded, uniformly positive material weight, the operator is invertible and supports the constitutive equations and energy expression.Source: §2.2 · p. 4
2.3 Spline complexes of differential forms
- Primal spline complexes use mixed degrees p and p−1, while the dual complexes use p−1 and p−2; the physical-domain forms are obtained by pullback.The spline derivative maps between the corresponding degree spaces, and the parameterization must have sufficient regularity for the approximation properties.Source: §2.3 · pp. 4–5
2.4 Pairing matrices
- Pairing matrices consist of integrals of wedge products between primal and dual basis forms and are invertible under the stated stability condition.They are metric-independent, can be factored in the parametric domain using tensor-product structure, and relate to incidence matrices representing discrete exterior derivatives.Source: §2.4 · pp. 5–6
3 Spatial discretization with energy preserving Hodge–star operators
The spatial discretization offers two discrete Hodge-star choices: a mass-matrix scheme equivalent to standard Galerkin methods and a pairing-matrix scheme equivalent to Petrov–Galerkin methods.
3 Spatial discretization with energy preserving Hodge–star operators
- Both discrete Hodge choices preserve electromagnetic energy and satisfy the two Gauss laws in the semi-discrete method.The pairing-matrix scheme is the paper’s main contribution; the discussion assumes time-invariant material parameters.Source: §3 · p. 6
3.1 Spatial discretization of Ampère-Maxwell and Faraday equations
The discrete Ampère–Maxwell and Faraday equations evolve the spline degrees of freedom, while the two Gauss laws follow exactly from the discrete exterior derivative property.
3.1 Spatial discretization of Ampère-Maxwell and Faraday equations
- The discrete Maxwell system uses the degrees-of-freedom vectors for the electric field, magnetic field, electric displacement, and magnetic induction, with a discrete current source.The source is assumed to satisfy the stated discrete divergence-free condition.Source: §3.1 · p. 6
- Only Ampère–Maxwell and Faraday need discretization because applying the exterior derivative twice vanishes, making both Gauss laws exact and conserving charges.The source is specified to satisfy the discrete divergence-free condition.Source: §3.1 · p. 6
- The discrete electromagnetic energy is represented using pairing matrices, and the spatial discretization is completed by approximating the Hodge star operators.Source: §3.1 · p. 6
3.2 Discrete Hodge star operators
The section defines two discrete Hodge-star choices that produce different schemes by mapping between primal and dual spline complexes. A third local operator is excluded because its dual-to-primal approximation properties and energy conservation are not assured.
3.2 Discrete Hodge star operators
- The paper presents two alternative discrete Hodge-star operators, particularizing the general construction in; the choice depends on whether equation (1.5) or (2.4) is discretized, yielding two different numerical schemes.Source: §3.2 · p. 6
- The first choice defines operators from primal 1-forms to dual 2-forms and from dual 1-forms to primal 2-forms.These discrete operators mimic the corresponding continuous Hodge-star definition.Source: §3.2 · pp. 6–7
- The alternative choice defines Hodge-star operators that map 2-forms into 1-forms.It likewise mimics the continuous Hodge-star definition, with the stated operator acting on dual 2-forms and mapping to primal 1-forms.Source: §3.2 · p. 7
- The authors discard a third local operator because its dual-to-primal mapping may not preserve approximation properties or guarantee electromagnetic-energy conservation.The concern is specifically with the operator studied in [32].Source: §3.2 · p. 7
3.3 First discretization scheme
The first discretization scheme imposes the time- and exterior-derivative equations strongly and the Hodge operators weakly, requiring mass-matrix solves. It conserves discrete energy and is equivalent to a mixed Galerkin formulation.
3.3 First discretization scheme
- The first scheme’s Hodge-star applications require pairing-matrix products and linear solves with mass matrices.The other two semi-discrete equations require only incidence-matrix applications.Source: §3.3 · p. 7
3.3.1 Energy conservation
- With zero source current and time-invariant permittivity and permeability, the first scheme conserves discrete energy.Factoring the mass matrices out of the time derivative yields conservation by standard arguments.Source: §3.3.1 · p. 8
3.3 First discretization scheme
- The four-equation formulation imposes time and exterior derivatives strongly while imposing the Hodge-star operators weakly.The authors compare this arrangement to the usual procedure in FIT formulations.Source: §3.3.2 · p. 8
3.3.2 Weak formulation in terms of E and H
- The equivalent mixed Galerkin formulation discretizes the electric field in the primal complex and the magnetic field in the dual complex.The authors describe this as similar to.Source: §3.3.2 · p. 8
3.4 Second discretization scheme
The second discretization scheme uses Hodge operators mapping 2-forms to 1-forms, so its Hodge-star solves involve pairing matrices rather than mass matrices. It conserves energy and is equivalent to a Petrov-Galerkin formulation.
3.4 Second discretization scheme
- The second scheme’s Hodge-star equations require linear solves with pairing matrices rather than mass matrices.The pairing matrices are generally not symmetric positive definite, but their tensor-product structure and metric independence offer advantages.Source: §3.4 · p. 9
3.4.1 Energy conservation
- The second scheme conserves energy, with the conservation argument relying on symmetric positive-definite mass matrices.Its energy can also be expressed as a quadratic form based on mass matrices for discrete 2-forms.Source: §3.4.1 · p. 9
3.4.2 Weak formulation in terms of E and H
- The equivalent formulation is Petrov-Galerkin, with electric and magnetic fields as 1-forms in the primal and dual complexes, respectively.The test-function spaces are 2-form spaces from the opposite complexes.Source: §3.4.2 · p. 10
- For most dual-grid methods, applying the Hodge operator entails solving a lumped-mass system; diagonal system matrices have not, to the authors’ knowledge, been extended to high-order Maxwell bases.The cited identity between pairing and identity matrices holds only for particular dual-mesh choices.Source: §3.4.2 · p. 10
3.5 Generalization to non-homogeneous boundary conditions
Non-homogeneous electric-field boundary data are handled by splitting the field into a homogeneous-trace component and a lifting. The lifting enters the right-hand side, while boundary-inclusive matrices are applied but not inverted.
3.5 Generalization to non-homogeneous boundary conditions
- The electric field is split into a component with vanishing tangential trace and a suitable boundary lifting computed by local projection.The projection uses basis functions with non-vanishing boundary trace.Source: §3.5 · p. 10
- The lifting contributes to the right-hand side after the electric-field expression is substituted into the scheme.The text introduces a block structure for the mass matrix and then applies the first scheme’s discrete Hodge star.Source: §3.5 · p. 10
- Boundary basis functions are included in the incidence and Hodge-related matrices, making some matrices rectangular; this is acceptable because they are applied, not inverted.For the second scheme, the internal block of the pairing matrix remains square and invertible for the Hodge-star solve.Source: §3.5 · pp. 10–11
4 Fast inversion of pairing matrices with Kronecker product structure
The section exploits the tensor-product structure of spline pairing matrices to solve the proposed systems without forming full Kronecker matrices. Its complexity analysis compares this approach with solving mass-matrix systems.
4.1 Structure of the pairing matrices
- Pairing matrices are independent of domain geometry and material parameters; Cartesian separability makes them block diagonal, with blocks that can be inverted separately and in parallel.Each block can be computed as a Kronecker product of univariate pairing matrices; the univariate pairings are invertible under the stated inf-sup condition.Source: §4 · p. 11, §4.1 · pp. 11–12
4 Fast inversion of pairing matrices with Kronecker product structure
- The proposed scheme solves systems involving pairing matrices rather than mass matrices, and these pairing matrices arise from discrete wedge-product integrals.The pairing matrices can be computed in the parametric domain.Source: §4 · p. 11
4.2 Solution of the linear system
- Kronecker inverse and vectorization identities reduce each system to smaller univariate solves, without explicitly forming the full Kronecker matrix.The rightmost system still requires multiple right-hand sides, so the tensor-product structure cannot be exploited further to avoid forming that intermediate Kronecker matrix.Source: §4.2 · pp. 12–13
4.3 Computational complexity
- The Kronecker scheme costs O(Nn^3p^3), versus O(Nn^6p^3) for mass-matrix solves; its per-step linear-system solve costs O(n^3p^2).Without exploiting tensor-product structure, factorization costs O(n^6p^3).Source: §4.3 · pp. 13–14
5 Numerical results
The numerical tests show energy conservation and degree-dependent field convergence, while the Kronecker-structured solver substantially reduces per-step cost. The coaxial-cable test retains the cavity problem’s convergence rates.
5 Numerical results
- The tests use maximum-regularity splines and a low-order leapfrog integrator, implemented in MATLAB with GeoPDEs.Computational-cost comparisons include direct MATLAB solves and solves exploiting Kronecker structure.Source: §5 · p. 14, §5.2 · p. 18
5.1.1 Cavity problem in the unit cube
- Energy is conserved up to leapfrog oscillations, whose amplitude and error relative to exact energy decrease with higher degree and mesh refinement.The cavity test uses p = 3, 4 and CFL-based time steps.Source: Figure 1 · p. 15, §5.1.1 · p. 15
- In the unit cube, electric-field error converges as h^p and magnetic-field error as h^(p−1); both schemes have similar error magnitudes.The finest-level electric-field result for p = 4 plateaus because of time-discretization error.Source: Figure 2 · p. 16, §5.1.1 · p. 16
5.1.2 Coaxial cable
- The coaxial-cable test reproduces the unit-cube convergence rates, with similar error magnitudes and no clear advantage for either scheme.The test simulates the TEM mode on a quarter of the cable geometry.Source: §5.1.2 · pp. 16–17, Figure 4 · p. 17
5.2 Study of the computational cost
- The pairing-matrix scheme’s CFL step scales as O(1/p^2), versus O(1/p^(3/2)) for the mass-matrix scheme; both scale linearly with mesh size.The mass-matrix scheme has some advantage in maximum time step with respect to mesh size.Source: §5.2 · pp. 17–18, Figure 5 · p. 18
- Roughly two orders of magnitude lower per-step cost comes from tensorized pairing solves, with gains increasing on finer meshes.For p = 4 on the finest coaxial-cable mesh, the per-step time was around 80 times lower; combining this with the CFL results gave a rough estimate of 40 times lower overall time.Source: §5.2 · pp. 18–19, Figure 6 · p. 19, Figure 7 · p. 19
6 Conclusions
The paper develops a high-degree spline method for Maxwell equations using two de Rham complexes and discrete Hodge stars. Its pairing matrices enable Kronecker-structured solves that lower per-step cost while maintaining accuracy and conservation properties.
6 Conclusions
- The method discretizes Maxwell equations with high-degree splines, two de Rham complexes, and specially constructed discrete Hodge star operators.The discrete Hodge stars require solving linear systems for pairing matrices with Kronecker tensor-product structure.Source: §6 · p. 20
- Exploiting the pairing matrices’ Kronecker structure reduces per-step computational cost by two orders of magnitude compared with solving mass-matrix systems, while maintaining discretization accuracy.Source: §6 · p. 20
- The method preserves Maxwell’s geometric structure, including charge and energy conservation; future work includes higher-order symplectic integrators and multipatch Kronecker inversion.Source: §6 · p. 20