Results for “pseudospectral method” — Page 6

NASA NTRS1985

Recovering pointwise values of discontinuous data within spectral accuracy

The pointwise values of a function, f(x), can be accurately recovered either from its spectral or pseudospectral approximations, so that the accuracy solely depends on the local smoothness of f in the neighborhood of the point x. Most notably, given the equidistant function grid values, its intermediate point values are recovered within spectral accuracy, despite the possible presence of discontinuities scattered in the domain. (Recall that the usual spectral convergence rate decelerates otherwise to first order, throughout). To this end, a highly oscillatory smoothing kernel is employed in contrast to the more standard positive unit-mass mollifiers. In particular, post-processing of a stable Fourier method applied to hyperbolic equations with discontinuous data, recovers the exact solution modulo a spectrally small error. Numerical examples are presented.

Gottlieb, D.↗

NASA NTRS1985

Recovering pointwise values of discontinuous data within spectral accuracy

The pointwise values of a function, f(x), can be accurately recovered either from its spectral or pseudospectral approximations, so that the accuracy solely depends on the local smoothness of f in the neighborhood of the point x. Most notably, given the equidistant function grid values, its intermediate point values are recovered within spectral accuracy, despite the possible presence of discontinuities scattered in the domain. (Recall that the usual spectral convergence rate decelerates otherwise to first order, throughout). To this end, a highly oscillatory smoothing kernel is employed in contrast to the more standard positive unit-mass mollifiers. In particular, post-processing of a stable Fourier method applied to hyperbolic equations with discontinuous data, recovers the exact solution modulo a spectrally small error. Numerical examples are presented.

Gottlieb, D.↗

DOE OSTI2025

Towards Automated Reasoning Chains for Verification of LLM-Generated Scientific Code

With the rise of Large Language Model (LLM) generated code, including in domains like scientific computing, ensuring not only syntactical, but also mathematical correctness, has become a critical task. Traditional formal methods approaches often struggle with the ambiguity of floating-point code, and full symbolic execution is extremely costly and limited. We propose a chain-of-reasoning approach that iteratively lifts basic semantics from code into the SPIRAL system and then establishes numerical equivalency to the desired mathematical operation. Here, we leverage the ample mathematical knowledge already formalized in SPIRAL to enable the system to recognize not just different implementations of the same algorithm but fully separate approaches to solving the given problem. The chain establishes tight error bounds on the output of given code with respect to the true continuous solution it approximates, quantifying all sources of error. We demonstrate this approach by establishing the correctness of a pseudospectral solver for a simple 1-dimensional Poisson problem.

Oschatz, Quentin [Carnegie Mellon University,Pitts↗

DOE OSTI2025

Accelerating magnonic simulations with the pseudospectral Landau-Lifshitz equation

The pseudospectral Landau-Lifshitz (PS-LL) model can describe atomic-scale magnetic exchange interactions within a continuum framework. This is achieved by employing a convolution kernel that models the nonlocal interaction in a grid-independent manner. Even though the PS-LL was originally introduced to address atomic exchange, any nonlocal kernel can be modeled. In the field of magnonics, the dipole field is fundamental to describe the dispersion relation of magnons, the quasiparticle representation of angular momentum. Because dipole-dipole interactions are long-range, numerical approaches typically rely on convolutions. Here, we demonstrate that the PS-LL model can be used to perform magnonic simulations with a single convolution kernel derived from analytical solutions. We demonstrate a twofold increase in computational speed compared with the full dipole calculation. This approach is valid insofar as the excitations are linear, which is typically the case for magnons. Our results have the potential to accelerate magnonic research, particularly for the inverse design method, where several simulations must be performed to achieve the desired outcome.

Mathematics and computing↗