We introduce a data-assisted, split Picard solver that cuts both iteration count and per-iteration cost for steady incompressible flows, enabling robust high-Re convergence and providing strong initial guesses for Newton methods.
Recommended citation: V. L. Fisher, L. G. Rebholz, & D. Vargun. "An efficient splitting iteration for a CDA-accelerated solver for incompressible flow problems". arXiv preprint, arXiv:2509.12547.(2025) Download Paper
We couple KORC and NIMROD to simulate post-disruption runaway electrons with robust particle-to-mesh mapping. We validate deposition, resolve near-axis accuracy needs, and introduce a low-cost orbit-averaging method that reduces mesh-induced statistical noise—laying groundwork for future self-consistent coupling.
Recommended citation: O. E. López, D. Vargun, C. D. Hauck, M. T. Beidler. "Orbit-averaging and deposition accuracy for runaway electron beams in hybrid kinetic-MHD simulations of the runaway plateau". Phys. Plasmas 1 August 2025, 32 (8):083908. Download Paper
Published in arxiv preprint - accepted for publications in Advances in Computational Mathematics, 2025
We analyze a fixed-point solver for the incompressible Oldroyd-B system and show that Anderson acceleration provably improves its linear convergence and extends robustness to higher Weissenberg numbers, confirmed by benchmark tests.
Recommended citation: D. Vargun, I. O. Monteiro, and L. G. Rebholz. "Anderson acceleration of a Picard solver for the Oldroyd-B model of viscoelastic fluids". arXiv preprint, arXiv:2502.00533 (2025). Download Paper
Published in Computer Methods in Applied Mechanics and Engineering, 2023
We augment Picard and Newton Navier–Stokes solvers with CDA measurement nudging. The result is faster convergence and improved robustness—CDA-Picard stays contractive at higher Reynolds numbers, and CDA-Newton’s convergence basin grows as more data is incorporated.
Recommended citation: X. Li, E. Hawkins, L. G. Rebholz, D. Vargun. "Accelerating and enabling convergence of nonlinear solvers for Navier–Stokes equations by continuous data assimilation". Computer Methods in Applied Mechanics and Engineering, 416:116313, 2023. Download Paper
Published in Communications in Mathematical Research, 2023
We apply continuous data assimilation to projection and penalty Navier–Stokes methods. With measurement data and proper parameter choices, CDA mitigates penalty modeling error and projection splitting error, yielding efficient schemes with long-time, optimally accurate solutions.
Recommended citation: E. Hawkins, L.G. Rebholz, D. Vargun. "Removing splitting/modeling error in projection/penalty methods for Navier-Stokes simulations with continuous data assimilation", Communications in Mathematical Research, 40(1):1–29, 2024. doi:10.4208/cmr.2023-0008 Download Paper
Published in Numerical Methods for Partial Differential Equations, 2023
We develop a finite element method for regularized Bingham flow and an efficient AA-accelerated Picard solver with proven faster convergence and robust performance as the regularization parameter goes to zero, validated in 2D/3D driven cavity tests.
Recommended citation: S. Pollock, L. G. Rebholz, and D. Vargun. "Anderson acceleration for a regularized Bingham model." Numerical Methods for Partial Differential Equations 39.5 (2023):3874-3896. Download Paper
Published in Journal of Computational and Applied Mathematics, 2023
We accelerate the classical Arrow–Hurwicz iteration for steady incompressible Navier–Stokes by adding grad–div stabilization and Anderson acceleration, and show analytically and numerically that the combined approach significantly improves convergence and is competitive with standard solvers.
Recommended citation: P. G. Geredeli, L. G. Rebholz, D. Vargun, and A. Zytoon. "Improved convergence of the Arrow–Hurwicz iteration for the Navier–Stokes equation via grad–div stabilization and Anderson acceleration." Journal of Computational and Applied Mathematics 422 (2023):114920. Download Paper
Published in Computer Methods in Applied Mechanics and Engineering, 2021
We show that Anderson acceleration improves the iterated penalty Picard solver for incompressible Navier–Stokes with an O(1) penalty parameter by leveraging Lipschitz smoothness of the fixed-point map, yielding provably faster linear convergence and excellent benchmark performance even with penalty parameter equal to 1.
Recommended citation: L. G. Rebholz, D. Vargun, and M. Xiao. "Enabling convergence of the iterated penalty Picard iteration with O (1) penalty parameter for incompressible Navier–Stokes via Anderson acceleration." Computer Methods in Applied Mechanics and Engineering 387 (2021):114178. Download Paper
We analyze CDA for a 2D velocity–vorticity Navier–Stokes method, proving that nudging (velocity-only or velocity-plus-vorticity) preserves unconditional long-time stability and yields optimal long-time accuracy, with vorticity nudging accelerating the approach to optimal accuracy and benchmarks confirming effectiveness for channel flow past a flat plate.
Recommended citation: M. Gardner, A. Larios, L. G. Rebholz, D. Vargun, and C. Zerfas. "Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations." Electronic Research Archive 29, no. 3 (2021). Download Paper
We introduce a data-assisted, split Picard solver that cuts both iteration count and per-iteration cost for steady incompressible flows, enabling robust high-Re convergence and providing strong initial guesses for Newton methods.
Recommended citation: V. L. Fisher, L. G. Rebholz, & D. Vargun. "An efficient splitting iteration for a CDA-accelerated solver for incompressible flow problems". arXiv preprint, arXiv:2509.12547.(2025) Download Paper
Published in Computer Methods in Applied Mechanics and Engineering, 2023
We augment Picard and Newton Navier–Stokes solvers with CDA measurement nudging. The result is faster convergence and improved robustness—CDA-Picard stays contractive at higher Reynolds numbers, and CDA-Newton’s convergence basin grows as more data is incorporated.
Recommended citation: X. Li, E. Hawkins, L. G. Rebholz, D. Vargun. "Accelerating and enabling convergence of nonlinear solvers for Navier–Stokes equations by continuous data assimilation". Computer Methods in Applied Mechanics and Engineering, 416:116313, 2023. Download Paper
Published in Communications in Mathematical Research, 2023
We apply continuous data assimilation to projection and penalty Navier–Stokes methods. With measurement data and proper parameter choices, CDA mitigates penalty modeling error and projection splitting error, yielding efficient schemes with long-time, optimally accurate solutions.
Recommended citation: E. Hawkins, L.G. Rebholz, D. Vargun. "Removing splitting/modeling error in projection/penalty methods for Navier-Stokes simulations with continuous data assimilation", Communications in Mathematical Research, 40(1):1–29, 2024. doi:10.4208/cmr.2023-0008 Download Paper
We analyze CDA for a 2D velocity–vorticity Navier–Stokes method, proving that nudging (velocity-only or velocity-plus-vorticity) preserves unconditional long-time stability and yields optimal long-time accuracy, with vorticity nudging accelerating the approach to optimal accuracy and benchmarks confirming effectiveness for channel flow past a flat plate.
Recommended citation: M. Gardner, A. Larios, L. G. Rebholz, D. Vargun, and C. Zerfas. "Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations." Electronic Research Archive 29, no. 3 (2021). Download Paper
We introduce a data-assisted, split Picard solver that cuts both iteration count and per-iteration cost for steady incompressible flows, enabling robust high-Re convergence and providing strong initial guesses for Newton methods.
Recommended citation: V. L. Fisher, L. G. Rebholz, & D. Vargun. "An efficient splitting iteration for a CDA-accelerated solver for incompressible flow problems". arXiv preprint, arXiv:2509.12547.(2025) Download Paper
Published in Computer Methods in Applied Mechanics and Engineering, 2023
We augment Picard and Newton Navier–Stokes solvers with CDA measurement nudging. The result is faster convergence and improved robustness—CDA-Picard stays contractive at higher Reynolds numbers, and CDA-Newton’s convergence basin grows as more data is incorporated.
Recommended citation: X. Li, E. Hawkins, L. G. Rebholz, D. Vargun. "Accelerating and enabling convergence of nonlinear solvers for Navier–Stokes equations by continuous data assimilation". Computer Methods in Applied Mechanics and Engineering, 416:116313, 2023. Download Paper
Published in Communications in Mathematical Research, 2023
We apply continuous data assimilation to projection and penalty Navier–Stokes methods. With measurement data and proper parameter choices, CDA mitigates penalty modeling error and projection splitting error, yielding efficient schemes with long-time, optimally accurate solutions.
Recommended citation: E. Hawkins, L.G. Rebholz, D. Vargun. "Removing splitting/modeling error in projection/penalty methods for Navier-Stokes simulations with continuous data assimilation", Communications in Mathematical Research, 40(1):1–29, 2024. doi:10.4208/cmr.2023-0008 Download Paper
Published in Journal of Computational and Applied Mathematics, 2023
We accelerate the classical Arrow–Hurwicz iteration for steady incompressible Navier–Stokes by adding grad–div stabilization and Anderson acceleration, and show analytically and numerically that the combined approach significantly improves convergence and is competitive with standard solvers.
Recommended citation: P. G. Geredeli, L. G. Rebholz, D. Vargun, and A. Zytoon. "Improved convergence of the Arrow–Hurwicz iteration for the Navier–Stokes equation via grad–div stabilization and Anderson acceleration." Journal of Computational and Applied Mathematics 422 (2023):114920. Download Paper
Published in Computer Methods in Applied Mechanics and Engineering, 2021
We show that Anderson acceleration improves the iterated penalty Picard solver for incompressible Navier–Stokes with an O(1) penalty parameter by leveraging Lipschitz smoothness of the fixed-point map, yielding provably faster linear convergence and excellent benchmark performance even with penalty parameter equal to 1.
Recommended citation: L. G. Rebholz, D. Vargun, and M. Xiao. "Enabling convergence of the iterated penalty Picard iteration with O (1) penalty parameter for incompressible Navier–Stokes via Anderson acceleration." Computer Methods in Applied Mechanics and Engineering 387 (2021):114178. Download Paper
We analyze CDA for a 2D velocity–vorticity Navier–Stokes method, proving that nudging (velocity-only or velocity-plus-vorticity) preserves unconditional long-time stability and yields optimal long-time accuracy, with vorticity nudging accelerating the approach to optimal accuracy and benchmarks confirming effectiveness for channel flow past a flat plate.
Recommended citation: M. Gardner, A. Larios, L. G. Rebholz, D. Vargun, and C. Zerfas. "Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations." Electronic Research Archive 29, no. 3 (2021). Download Paper
We couple KORC and NIMROD to simulate post-disruption runaway electrons with robust particle-to-mesh mapping. We validate deposition, resolve near-axis accuracy needs, and introduce a low-cost orbit-averaging method that reduces mesh-induced statistical noise—laying groundwork for future self-consistent coupling.
Recommended citation: O. E. López, D. Vargun, C. D. Hauck, M. T. Beidler. "Orbit-averaging and deposition accuracy for runaway electron beams in hybrid kinetic-MHD simulations of the runaway plateau". Phys. Plasmas 1 August 2025, 32 (8):083908. Download Paper
We couple KORC and NIMROD to simulate post-disruption runaway electrons with robust particle-to-mesh mapping. We validate deposition, resolve near-axis accuracy needs, and introduce a low-cost orbit-averaging method that reduces mesh-induced statistical noise—laying groundwork for future self-consistent coupling.
Recommended citation: O. E. López, D. Vargun, C. D. Hauck, M. T. Beidler. "Orbit-averaging and deposition accuracy for runaway electron beams in hybrid kinetic-MHD simulations of the runaway plateau". Phys. Plasmas 1 August 2025, 32 (8):083908. Download Paper
We couple KORC and NIMROD to simulate post-disruption runaway electrons with robust particle-to-mesh mapping. We validate deposition, resolve near-axis accuracy needs, and introduce a low-cost orbit-averaging method that reduces mesh-induced statistical noise—laying groundwork for future self-consistent coupling.
Recommended citation: O. E. López, D. Vargun, C. D. Hauck, M. T. Beidler. "Orbit-averaging and deposition accuracy for runaway electron beams in hybrid kinetic-MHD simulations of the runaway plateau". Phys. Plasmas 1 August 2025, 32 (8):083908. Download Paper
We couple KORC and NIMROD to simulate post-disruption runaway electrons with robust particle-to-mesh mapping. We validate deposition, resolve near-axis accuracy needs, and introduce a low-cost orbit-averaging method that reduces mesh-induced statistical noise—laying groundwork for future self-consistent coupling.
Recommended citation: O. E. López, D. Vargun, C. D. Hauck, M. T. Beidler. "Orbit-averaging and deposition accuracy for runaway electron beams in hybrid kinetic-MHD simulations of the runaway plateau". Phys. Plasmas 1 August 2025, 32 (8):083908. Download Paper
Published in arxiv preprint - accepted for publications in Advances in Computational Mathematics, 2025
We analyze a fixed-point solver for the incompressible Oldroyd-B system and show that Anderson acceleration provably improves its linear convergence and extends robustness to higher Weissenberg numbers, confirmed by benchmark tests.
Recommended citation: D. Vargun, I. O. Monteiro, and L. G. Rebholz. "Anderson acceleration of a Picard solver for the Oldroyd-B model of viscoelastic fluids". arXiv preprint, arXiv:2502.00533 (2025). Download Paper
Published in Numerical Methods for Partial Differential Equations, 2023
We develop a finite element method for regularized Bingham flow and an efficient AA-accelerated Picard solver with proven faster convergence and robust performance as the regularization parameter goes to zero, validated in 2D/3D driven cavity tests.
Recommended citation: S. Pollock, L. G. Rebholz, and D. Vargun. "Anderson acceleration for a regularized Bingham model." Numerical Methods for Partial Differential Equations 39.5 (2023):3874-3896. Download Paper
Published in Journal of Computational and Applied Mathematics, 2023
We accelerate the classical Arrow–Hurwicz iteration for steady incompressible Navier–Stokes by adding grad–div stabilization and Anderson acceleration, and show analytically and numerically that the combined approach significantly improves convergence and is competitive with standard solvers.
Recommended citation: P. G. Geredeli, L. G. Rebholz, D. Vargun, and A. Zytoon. "Improved convergence of the Arrow–Hurwicz iteration for the Navier–Stokes equation via grad–div stabilization and Anderson acceleration." Journal of Computational and Applied Mathematics 422 (2023):114920. Download Paper
Published in Computer Methods in Applied Mechanics and Engineering, 2021
We show that Anderson acceleration improves the iterated penalty Picard solver for incompressible Navier–Stokes with an O(1) penalty parameter by leveraging Lipschitz smoothness of the fixed-point map, yielding provably faster linear convergence and excellent benchmark performance even with penalty parameter equal to 1.
Recommended citation: L. G. Rebholz, D. Vargun, and M. Xiao. "Enabling convergence of the iterated penalty Picard iteration with O (1) penalty parameter for incompressible Navier–Stokes via Anderson acceleration." Computer Methods in Applied Mechanics and Engineering 387 (2021):114178. Download Paper
Published in Numerical Methods for Partial Differential Equations, 2023
We develop a finite element method for regularized Bingham flow and an efficient AA-accelerated Picard solver with proven faster convergence and robust performance as the regularization parameter goes to zero, validated in 2D/3D driven cavity tests.
Recommended citation: S. Pollock, L. G. Rebholz, and D. Vargun. "Anderson acceleration for a regularized Bingham model." Numerical Methods for Partial Differential Equations 39.5 (2023):3874-3896. Download Paper
Published in arxiv preprint - accepted for publications in Advances in Computational Mathematics, 2025
We analyze a fixed-point solver for the incompressible Oldroyd-B system and show that Anderson acceleration provably improves its linear convergence and extends robustness to higher Weissenberg numbers, confirmed by benchmark tests.
Recommended citation: D. Vargun, I. O. Monteiro, and L. G. Rebholz. "Anderson acceleration of a Picard solver for the Oldroyd-B model of viscoelastic fluids". arXiv preprint, arXiv:2502.00533 (2025). Download Paper
Published in Journal of Computational and Applied Mathematics, 2023
We accelerate the classical Arrow–Hurwicz iteration for steady incompressible Navier–Stokes by adding grad–div stabilization and Anderson acceleration, and show analytically and numerically that the combined approach significantly improves convergence and is competitive with standard solvers.
Recommended citation: P. G. Geredeli, L. G. Rebholz, D. Vargun, and A. Zytoon. "Improved convergence of the Arrow–Hurwicz iteration for the Navier–Stokes equation via grad–div stabilization and Anderson acceleration." Journal of Computational and Applied Mathematics 422 (2023):114920. Download Paper
Published in Computer Methods in Applied Mechanics and Engineering, 2021
We show that Anderson acceleration improves the iterated penalty Picard solver for incompressible Navier–Stokes with an O(1) penalty parameter by leveraging Lipschitz smoothness of the fixed-point map, yielding provably faster linear convergence and excellent benchmark performance even with penalty parameter equal to 1.
Recommended citation: L. G. Rebholz, D. Vargun, and M. Xiao. "Enabling convergence of the iterated penalty Picard iteration with O (1) penalty parameter for incompressible Navier–Stokes via Anderson acceleration." Computer Methods in Applied Mechanics and Engineering 387 (2021):114178. Download Paper
