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Reduced Augmentation Implicit Low-rank (RAIL) integrators for advection-diffusion and Fokker-Planck models

Joseph Nakao, Jing-Mei Qiu, Lukas Einkemmer

TL;DR

RAIL blends dynamical low-rank and step-and-trunc ideas to solve time-dependent PDEs by spectral spatial discretization and implicit or IMEX Runge-Kutta time stepping. It updates low-rank bases at each RK stage via a reduced augmentation that combines predictions with past bases, followed by SVD truncation to control rank and a mass-conserving post-processing step. The method demonstrates high-order temporal accuracy, efficient low-rank representation through Sylvester-type projections, and robust global mass conservation in advection-diffusion and Fokker-Planck tests. Overall, RAIL provides a unified, efficient framework that bridges DLR and SAT approaches and holds promise for scalable, structure-preserving solvers in higher dimensions and for broader low-rank tensor decompositions.

Abstract

This paper introduces a novel computational approach termed the Reduced Augmentation Implicit Low-rank (RAIL) method by investigating two predominant research directions in low-rank solutions to time-dependent partial differential equations (PDEs): dynamical low-rank (DLR), and step and truncation (SAT) tensor methods. The RAIL method, along with the development of the SAT approach, is designed to enhance the efficiency of traditional full-rank implicit solvers from method-of-lines discretizations of time-dependent PDEs, while maintaining accuracy and stability. We consider spectral methods for spatial discretization, and diagonally implicit Runge-Kutta (DIRK) and implicit-explicit (IMEX) RK methods for time discretization. The efficiency gain is achieved by investigating low-rank structures within solutions at each RK stage using a singular value decomposition (SVD). In particular, we develop a reduced augmentation procedure to predict the basis functions to construct projection subspaces. This procedure balances algorithm accuracy and efficiency by incorporating as many bases as possible from previous RK stages and predictions, and by optimizing the basis representation through SVD truncation. As such, one can form implicit schemes for updating basis functions in a dimension-by-dimension manner, similar in spirit to the K-L step in the DLR framework. We also apply a globally mass conservative post-processing step at the end of each RK stage. We validate the RAIL method through numerical simulations of advection-diffusion problems and a Fokker-Planck model, showcasing its ability to efficiently handle time-dependent PDEs while maintaining global mass conservation. Our approach generalizes and bridges the DLR and SAT approaches, offering a comprehensive framework for efficiently and accurately solving time-dependent PDEs with implicit treatment.

Reduced Augmentation Implicit Low-rank (RAIL) integrators for advection-diffusion and Fokker-Planck models

TL;DR

RAIL blends dynamical low-rank and step-and-trunc ideas to solve time-dependent PDEs by spectral spatial discretization and implicit or IMEX Runge-Kutta time stepping. It updates low-rank bases at each RK stage via a reduced augmentation that combines predictions with past bases, followed by SVD truncation to control rank and a mass-conserving post-processing step. The method demonstrates high-order temporal accuracy, efficient low-rank representation through Sylvester-type projections, and robust global mass conservation in advection-diffusion and Fokker-Planck tests. Overall, RAIL provides a unified, efficient framework that bridges DLR and SAT approaches and holds promise for scalable, structure-preserving solvers in higher dimensions and for broader low-rank tensor decompositions.

Abstract

This paper introduces a novel computational approach termed the Reduced Augmentation Implicit Low-rank (RAIL) method by investigating two predominant research directions in low-rank solutions to time-dependent partial differential equations (PDEs): dynamical low-rank (DLR), and step and truncation (SAT) tensor methods. The RAIL method, along with the development of the SAT approach, is designed to enhance the efficiency of traditional full-rank implicit solvers from method-of-lines discretizations of time-dependent PDEs, while maintaining accuracy and stability. We consider spectral methods for spatial discretization, and diagonally implicit Runge-Kutta (DIRK) and implicit-explicit (IMEX) RK methods for time discretization. The efficiency gain is achieved by investigating low-rank structures within solutions at each RK stage using a singular value decomposition (SVD). In particular, we develop a reduced augmentation procedure to predict the basis functions to construct projection subspaces. This procedure balances algorithm accuracy and efficiency by incorporating as many bases as possible from previous RK stages and predictions, and by optimizing the basis representation through SVD truncation. As such, one can form implicit schemes for updating basis functions in a dimension-by-dimension manner, similar in spirit to the K-L step in the DLR framework. We also apply a globally mass conservative post-processing step at the end of each RK stage. We validate the RAIL method through numerical simulations of advection-diffusion problems and a Fokker-Planck model, showcasing its ability to efficiently handle time-dependent PDEs while maintaining global mass conservation. Our approach generalizes and bridges the DLR and SAT approaches, offering a comprehensive framework for efficiently and accurately solving time-dependent PDEs with implicit treatment.
Paper Structure (19 sections, 2 theorems, 66 equations, 6 figures, 6 tables, 3 algorithms)

This paper contains 19 sections, 2 theorems, 66 equations, 6 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. The matrix differential equation arising from PDE discretization in space
  3. The implicit RAIL scheme
  4. The first-order implicit scheme
  5. The second-order implicit scheme
  6. High-order implicit schemes
  7. Analysis of the implicit scheme
  8. Consistency
  9. Stability
  10. Extension to implicit-explicit (IMEX) time discretizations
  11. Numerical tests
  12. The diffusion equation
  13. Rigid body rotation with diffusion
  14. Swirling deformation with diffusion
  15. 0D2V Lenard-Bernstein-Fokker-Planck equation
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\mathbf{U}(t)$ denote the solution to the matrix differential equation eq:MOL_implicit with initial condition $\mathbf{U}(t^0) = \mathbf{U}^0_e$. Let $\mathbf{F}(t,\mathbf{U}(t))=\mathbf{F}_x\mathbf{U}+\mathbf{U}\mathbf{F}_y^T$. Assume the following conditions hold in the Frobenius norm $\norm{ Let $\mathbf{U}^1$ denote the low-rank approximation to $\mathbf{U}(t^1)$ obtained after one step o

Figures (6)

  • Figure 1: (left) Error plot for \ref{['eq:test_diffusion']} with initial condition \ref{['eq:diffusion_IC']} using the first-, second- and third-order RAIL schemes; mesh size $N=200$, tolerance $\epsilon=1.0E-08$, final time $T_f=0.5$, initial rank $r^0=20$. The rank of the solution (middle) and relative number density (right) to \ref{['eq:test_diffusion']} with initial condition \ref{['eq:diffusion_IC']}; mesh size $N=400$, tolerance $\epsilon=1.0E-08$, time-stepping size $\Delta t=0.3\Delta x$, initial rank $r^0=40$.
  • Figure 2: (left) Error plot for \ref{['eq:test_rigid']} with initial condition $\text{exp}(-(x^2+3y^2+2d t))$ using the first-, second- and third-order RAIL schemes; mesh size $N=200$, tolerance $\epsilon=1.0E-08$, final time $T_f=0.5$, initial rank $r^0=20$. The rank of the solution (middle) and relative number density (right) to \ref{['eq:test_rigid']} with initial condition $\text{exp}(-(x^2+9y^2))$; mesh size $N=200$, tolerance $\epsilon=1.0E-08$, time-stepping size $\Delta t=0.15\Delta x$, initial rank $r^0=20$.
  • Figure 3: Various snapshots of the numerical solution to equation \ref{['eq:test_rigid']} with initial condition $\text{exp}(-(x^2+9y^2))$. Mesh size $N=200$, tolerance $\epsilon=1.0E-08$, time-stepping size $\Delta t=0.15\Delta x$, initial rank $r^0=20$, using IMEX(4,4,3). Times: 0, $\pi/4$, $\pi/2$.
  • Figure 4: (left) Error plot for \ref{['eq:test_swirl']} with initial condition \ref{['eq:cosbell']} using the first-, second- and third-order RAIL schemes; mesh size $N=100$, tolerance $\epsilon=1.0E-08$, final time $T_f=0.5$, initial rank $r^0=15$. The rank of the solution (middle) and relative number density (right) to \ref{['eq:test_swirl']} with initial condition \ref{['eq:cosbell']}; mesh size $N=300$, tolerance $\epsilon=1.0E-08$, time-stepping size $\Delta t=0.15\Delta x$, initial rank $r^0=30$.
  • Figure 5: The rank of the solution (left) and relative number density (middle) to \ref{['eq:test_LBFP']} with initial condition $f_{M1}(v_x,v_y) + f_{M2}(v_x,v_y)$; mesh size $N=300$, tolerance $\epsilon=1.0E-06$, time-stepping size $\Delta t=0.15\Delta x$, initial rank $r^0=30$. (right) Error plot for \ref{['eq:test_LBFP']} with initial condition $f_{M1}(v_x,v_y) + f_{M2}(v_x,v_y)$; mesh size $N=300$, tolerance $\epsilon=1.0E-06$, final time $T_f=15$, $\lambda=0.15$.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Remark 1
  • Theorem 1: see Theorem 2 in Ceruti2022a, and Lemma 4 in Ceruti2022
  • Remark 2
  • Theorem 2
  • proof
  • Remark 3
  • Remark 4