We present a tensor-train discontinuous Galerkin (TT-DG) formulation for the Vlasov--Maxwell system that combines a modal DG discretization with low-rank tensor representations of the phase-space solution and discrete operators. The formulation exploits the tensor-product structure of the DG discretization to perform quadrature, differentiation, nonlinear upwind flux evaluation, and time integration directly in compressed form. The method is evaluated on several standard 1D2V Vlasov--Maxwell benchmark problems, including the streaming Weibel instability, weak Landau damping, and two-stream instability problems. Across these problems, the TT formulation reproduces the accuracy and conservation behavior of the underlying full-grid DG discretization while substantially reducing memory usage and runtime. For weakly nonlinear problems, compression ratios exceeding $10^4$ are obtained together with significant speedups relative to the full-grid solver. For the strongly nonlinear two-stream instability problem, the TT formulation remains effective despite reduced compressibility caused by fine-scale phase-space filamentation. These results demonstrate that tensor-train representations provide an effective approach for reducing the computational cost of deterministic DG-based kinetic plasma simulations while retaining the favorable numerical properties of the underlying discretization.
This work studies adaptive sparse-grid discontinuous Galerkin (DG) discretizations for the Bhatnagar--Gross--Krook (BGK) model, a kinetic equation posed in four- and six-dimensional phase-space. Standard DG methods are rendered impractical for the BGK model by the curse of dimensionality, motivating compressed representations that adapt to the solution in time. Using the adaptive sparse-grid DG method, we quantify accuracy and compression by comparing the adaptive degrees of freedom to full-grid DG methods and by assessing the resulting kinetic and fluid quantities in both fluid and rarefied regimes. Test cases include a relaxation problem, a multidimensional Sod shock tube, and shear/expansion flows used in prior low-rank BGK studies. To build an efficient Maxwellian evaluation without violating conservation, a central obstacle for structure-perserving BGK simulations, we introduce a hybrid interpolation strategy that exploits velocity separability to recover the correct discrete collision invariants and prove conservation of the resulting discrete collision operator on adaptive sparse grids. Our results show that the adaptive sparse-grid strategy can recover accurate and physically relevant solutions with sharp gradients, and the method reduces the active degrees of freedom by factors ranging from several-fold to several orders of magnitude, with the largest reductions occurring in the six-dimensional examples. All computations are performed with the open-source ASGarD adaptive sparse-grid DG library.
Stefan R. Schnake, M. Stoyanov, E. Endeve et al.· 0 citations
The Galerkin finite element formulation of the incompressible Navier-Stokes equations presents two principal challenges: maintaining stable velocity-pressure coupling and controlling instability in advection-dominated regimes. Moreover, the monolithic formulation produces a coupled nonlinear saddle-point problem. In this work, we present a residual-based variational multiscale (VMS) stabilization of an incremental Helmholtz-Leray projection method that replaces this coupled saddle-point problem with a nonlinear velocity predictor, a pressure Poisson equation, and a velocity projection. The multiscale decomposition is applied only to the predicted velocity; neither the pressure nor the corrected, weakly divergence-free velocity is decomposed into coarse and fine scales. The modeled velocity fine scale contributes consistently to all three subproblems, introducing SUPG-like stabilization in the momentum predictor and a PSPG-like residual contribution in the pressure Poisson equation. We provide a formal error decomposition that separates the BDF2 time-discretization, projection-splitting, and spatial-VMS errors and, under stated stability and spatial-approximation assumptions, yields a combined velocity error estimate with second-order temporal accuracy. Numerical results for manufactured solutions, lid-driven cavity flow, flow past a cylinder, and the Taylor-Green vortex agree closely with established reference data. Comparisons with monolithic VMS indicate that omitting the pressure fine scale reduces drag overprediction and excess modeled dissipation, at the cost of increased divergence error. In the Taylor-Green tests, the projection formulation also reduces the average solution time per step by factors ranging from approximately $1.3\times$ to $2.7\times$ under identical solver settings.
Biswajit Khara, Suresh Murugaiyan, Makrand A. Khanwale et al.· 0 citations
In this paper, we present a high-order, stable summation-by-parts (SBP) finite difference method for solving the Vlasov-Maxwell system in a 2D2V phase space. Central SBP operators for the advection terms are not stable when the solution becomes non-smooth and fine-scale filamentary structures develop, as is typical in high-dimensional Vlasov-Maxwell simulations. To address this issue, the method is stabilized using high-order upwind SBP operators. High-order explicit Runge-Kutta methods are employed for time integration. We prove that the fully discrete scheme exactly conserves mass and preserves momentum up to truncation error. Furthermore, we present a matrix-free implementation of the method on modern GPU architectures. A range of challenging benchmark problems is solved to demonstrate the accuracy, robustness, and performance of the proposed scheme.
Robin Dymér, K. Mattsson, Murtazo Nazarov· 0 citations
Time-fractional diffusion equations describe subdiffusive transport in heterogeneous media, but their numerical treatment is complicated by the nonlocal Caputo derivative and by the weak singularity that the solution develops at the initial time. We study a global-in-time discretization that combines spectral collocation in time—on the fractional power basis {tℓα}ℓ=0N, evaluated at Chebyshev–Gauss–Lobatto nodes, which reproduces the leading terms of the singular expansion of the solution—with a second-order conservative finite-difference stencil in space that uses harmonic averaging of the diffusivity at the cell faces and therefore remains accurate across discontinuous media. The resulting fully discrete problem is a large, nonsymmetric, dense-in-time linear system whose two-norm condition number grows like the inverse square of the spatial mesh size, so that Krylov subspace iteration without preconditioning stalls under refinement. Exploiting the Kronecker sum structure of the discrete operator, we build a preconditioner by fast diagonalization of the spatial factor through the discrete sine transform. For constant diffusivity the preconditioner reproduces the operator exactly and yields a direct solver; for variable diffusivity it is spectrally equivalent to the operator, and we prove that the eigenvalues of the preconditioned system cluster in a disk centered at one whose radius depends only on the coefficient contrast, and not on the mesh, the number of temporal degrees of freedom, or the fractional order. Numerical experiments in one and two space dimensions confirm second-order spatial accuracy and a preconditioned iteration count that stays flat—twelve iterations from M=32 up to M=1024 in one dimension and eleven up to M=256 per direction in two—while the unpreconditioned count grows by more than two orders of magnitude. In time, the accuracy is spectral until round-off in the ill-conditioned Vandermonde matrix of the power basis takes over: the barrier is reached at N=9,10,13 for α=0.3,0.5,0.7, where the attainable error is about 10−6. A benchmark against the L1 scheme on uniform and graded meshes, the Alikhanov L2-1σ scheme and Grünwald–Letnikov convolution quadrature quantifies when the global approach pays: on forced problems and on modes with κλTα≲2 it reaches a prescribed accuracy one to two orders of magnitude faster and with several times less memory, while for strongly damped modes the fractional power basis converges only algebraically and graded time marching is preferable below a relative error of 10−2.
This article presents a fast, high-order Nystr\"om solver for the two-dimensional Lippmann--Schwinger equation arising from time-harmonic scattering by penetrable, piecewise-smooth heterogeneous media. Relying on high-order evaluation of the Newtonian potential on unstructured grids adapted to interfaces of discontinuity, the methodology achieves high-order accuracy using existing fast algorithms such as the fast multipole method. As an iterative method the solver exhibits rapid convergence when coupled to a preconditioning strategy that exploits a class of structured-grid solvers---fast solution methods offering quasi-linear time and memory complexity and nearly-constant iteration counts, but long limited in accuracy. The preconditioning strategy couples the Nystr\"om discretization---given by high-order quadrature nodes over a (curved) unstructured mesh conforming to the support of the spatially varying contrast---to a uniform Cartesian grid underlying the fast preconditioner via a pair of transfer operators. The resulting preconditioner inherits the frequency-robust behavior of its Cartesian counterpart without sacrificing the geometric flexibility and high-order accuracy of the unstructured discretization. We prove that invertibility of the proposed preconditioner holds under explicit conditions on the mesh sizes and on the Cartesian preconditioner. Numerical experiments demonstrate that the preconditioned system requires significantly fewer GMRES iterations than its unpreconditioned counterpart, with iteration counts almost independent of mesh size and wavenumber, and illustrate the method's robustness for inhomogeneities with piecewise-smooth refractive indices and jump discontinuities across interfaces.
T. Anderson, Juan Burbano-Gallegos, Luiz M. Faria et al.· 0 citations
We develop a neural-network-accelerated multiscale hybridizable discontinuous Galerkin method for elliptic problems with heterogeneous coefficients. The method preserves the standard MsHDG local-to-global structure: fine-scale HDG problems on coarse blocks define discrete Dirichlet-to-Neumann operators, which are assembled through the standard MsHDG global skeleton equations. To reduce the cost of constructing these local operators, we train a neural network on coefficient fields defined on a reference block and use the predicted operators in place of repeated fine-scale local solves. The numerical experiments assess both the accuracy and online efficiency of the resulting NN-MsHDG method. For two-dimensional binary permeability fields with moderate contrast, the neural method reproduces the standard MsHDG solution with relative modeling errors of a few percent while reducing the total online computational cost by factors of approximately 5 to 16, depending on the coarse trace dimension. In the high-contrast regime, the chosen polynomial coarse trace spaces already yield substantial discretization errors in the standard MsHDG method, indicating the need for more effective coarse spaces, such as coefficient-adapted spectral trace spaces. In addition, the learned local operators introduce modeling errors that become severe as the trace space is enriched. These results demonstrate the potential of neural surrogates for accelerating multiscale HDG computations while also highlighting the need for improved operator representations and a better understanding of error amplification in high-contrast problems.
Tony Haines, Ke Shi· 0 citations
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