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A Deferred Correction, Continuous Galerkin Method for Curvilinear Staggered-Grid Lagrangian Hydrodynamics
Steven Walton, Svetlana Tokareva, Nathaniel Morgan
TL;DR
The paper addresses high-order continuous Galerkin Lagrangian hydrodynamics without global mass-matrix solves. It combines cG with explicit Deferred Correction and shock-compatible hyperviscosity, and reports accuracy comparable to cG-RK with efficiency gains on Taylor-Green tests for lower-order elements.
Problem
Continuous Galerkin Lagrangian methods require global sparse mass-matrix inversion, while strong shocks require stabilization compatible with a method that avoids that solve.
Source: §1 · pp. 2–3Method
The method combines high-order continuous finite elements with explicit Deferred Correction and modifies hyperviscosity to avoid a global solve for the polyharmonic approximation.
Source: §1 · pp. 2–3, §4 · p. 10Results
The method had comparable accuracy and better efficiency than cG-RK on Taylor-Green vortex tests with Q1Q0 and Q2Q1 elements, while extra corrections reduced efficiency above kinematic polynomial order two; Sedov solutions were robust.
Source: §6 · pp. 26–27Takeaways & Limitations
Preconditioning the DeC operators to reduce their spectral radius is identified as a possible route to requiring fewer corrections.
Source: §6 · p. 27The DeC convergence behavior depends on a spectral radius that can be close to one, depending on polynomial order, problem dimension, and basis type.
Source: §6 · p. 27
Abstract
from arXiv · showhide
We present a continuous Galerkin, Deferred Correction (cG-DeC) method for the equations of Lagrangian hydrodynamics. The proposed scheme combines a high-order continuous finite element discretization with an explicit Deferred Correction time integrator, yielding a formulation that avoids the inversion of a global sparse mass matrix at each update. To stabilize the method in the presence of strong shocks we introduce a modification of the hyperviscosity model that is compatible with the cG-DeC framework and likewise does not require a global solve to construct the finite element approximation to the polyharmonic operator. We further present an alternative reformulation of the DeC iteration which admits a simple recursive algorithmic structure and provides insight into previously observed convergence behavior of explicit DeC applied to hyperbolic partial differential equations. Conservation of momentum and total energy of the resulting fully discrete scheme is analyzed. A set of numerical experiments illustrates the accuracy and robustness of the cG-DeC scheme. Comparisons with a continuous Galerkin Runge-Kutta (cG-RK) method are provided.
1 Introduction
The paper applies cG-DeC to Lagrangian hydrodynamics to avoid global mass-matrix inversion, adapts hyperviscosity for the framework, and reformulates DeC to shed light on convergence behavior.
1 Introduction
- The paper positions continuous finite elements as an alternative to the finite-volume and discontinuous Galerkin methods prominent in hyperbolic conservation-law research.Prior high-order continuous methods had demonstrated effectiveness in Lagrangian simulations, while avoiding a multidimensional Riemann solver came with a global-solve cost.Source: §1 · p. 2
- Earlier staggered-grid Residual Distribution methods use DeC mass lumping to avoid a global mass-matrix solve, but require a distribution step for high order.The distribution step is needed because their residuals include a first-order viscosity term.Source: §1 · p. 2
- The modified hyperviscosity model constructs the polyharmonic finite element approximation without inverting a global sparse mass matrix.This makes the stabilization compatible with cG-DeC.Source: §1 · p. 3
- The alternative DeC iteration formulation sheds light on why theoretical correction counts may fail to deliver expected convergence order for hyperbolic PDEs.The theoretical iteration basis gives a fixed correction count for desired accuracy, but observed convergence may differ.Source: §1 · p. 3
- cG-DeC avoids the global sparse mass-matrix inversion required by continuous Galerkin momentum updates, a synchronization-intensive operation on exascale heterogeneous architectures.The cited discussion identifies global dot products and norm evaluations in Krylov solvers as communication-bound costs.Source: §1 · pp. 2–3
2 Governing Equations
The governing system is posed in a Lagrangian, co-moving frame and describes mass, momentum, and internal-energy evolution, with an ideal-gas equation of state.
2 Governing Equations
- The equations are formulated in a Lagrangian (co-moving) reference frame, with the material derivative used for time evolution.The spatial domain is considered in one, two, or three dimensions.Source: §2 · p. 3
- The system includes mass conservation, with density and velocity defined as fluid fields, and a momentum equation derived from Newton’s second law.The stress consists of pressure and artificial-viscosity contributions.Source: §2 · p. 3
- The specific internal-energy equation follows from the First Law of Thermodynamics and is complemented by an equation of motion.The equations are summarized as the system in Lagrangian coordinates.Source: §2 · p. 4
- The model assumes an ideal gas: internal energy is cvT and pressure is given by the ideal-gas equation of state.The specific heat capacity is constant, and γ is the adiabatic constant.Source: §2 · p. 4
3 Finite Element Discretization
The spatial discretization combines continuous kinematic and element-local thermodynamic finite element spaces with limited artificial viscosity, then expresses the resulting system in semi-discrete form.
3.1 Notation, Mesh Discretization and Spatial Operators
- Kinematic quantities use degree-q finite element polynomials, while thermodynamic quantities use degree-(q−1) polynomials on their reference elements.The stated basis choices are Lagrange on tensor-product elements and Bernstein on simplices.Source: §3.1 · pp. 4–5
- Kinematic degrees of freedom are continuous across the mesh, whereas thermodynamic degrees of freedom are not shared across elements.Element mappings connect reference coordinates to physical elements, and pointwise mass conservation determines density at quadrature points.Source: §3.1 · pp. 5–6
- Pressure and other thermodynamic quantities besides internal energy are evaluated at quadrature points, avoiding the need for hourglass controllers.The density follows pointwise mass conservation, and pressure is evaluated using the equation of state and interpolated internal energy.Source: §3.1 · p. 6
- The mass matrices are time independent and assembled once at the start of the simulation.This follows from the density relation and time-constant basis functions.Source: §3.1 · p. 6
3.2 Artificial Viscosity
- Limited artificial viscosity is designed to retain high-order accuracy in smooth regions while stabilizing shocks, using a Neumann-series approximation of the polyharmonic operator.The approximation uses a mesh-, basis-, and polynomial-order-dependent number of terms; one iteration sufficed for the reported tests.Source: §3.2 · pp. 7–8
- The limited viscosity coefficient is the minimum of hyperviscosity and standard viscosity, with the standard coefficient including linear and quadratic terms.The formulation also uses measures of vorticity and compression and a user-defined parameter for the quadratic term.Source: §3.2 · p. 7
3.3 The Semi-Discrete System
- The defined spatial operators are assembled into semi-discrete matrix-vector equations using a Picard integral formulation.Together with pointwise mass conservation, these equations complete the spatial discretization.Source: §3.3 · p. 8
4 Deferred Correction
The section develops an explicit, recursive Deferred Correction update for Lagrangian hydrodynamics and analyzes its convergence and fully discrete conservation properties.
4 Deferred Correction
- An equivalent closed-form DeC iteration reveals a Neumann-series approximation to the mass-matrix inverse, whose convergence depends on A's spectral radius.For high-order Bernstein polynomials, A's spectral radius approaches one, and the theoretical correction count may not yield the expected convergence.Source: §4 · pp. 11–12
- The proposed route to faster convergence—choosing an invertible L1 operator that better preconditions M—is conjectured but left beyond the article's scope.The motivation is to reduce A's spectral radius and speed up cG-DeC convergence.Source: §4 · p. 12
4.1 Application to the Equations of Lagrangian Hydrodynamics
- Applied to the Lagrangian equations, the recursive DeC algorithm reuses precomputed lumped mass and AV/AE matrices on correction residuals and stores only sub-time-node values.The paper notes that the storage requirement is similar to that of an explicit Runge-Kutta integrator.Source: §4.1 · p. 13
4.2 Conservation
- The fully discrete scheme conserves momentum, with the proof using partition of unity for the kinematic basis at every correction level.The proposition states the conservation result; the proof passage identifies the partition-of-unity property.Source: §4.2 · p. 15
4.2.1 Total Energy Conservation
- For the modified second-order DeC method, a final-iteration internal-energy correction conserves total energy independently of the number of corrections.The proposition states that applying the correction to the final internal-energy update conserves total energy.Source: §4.2.1 · p. 17
- Even when MV = DV, the high-order case still requires a further modification; under-integration of the force tensor impacts stability for d ≥ 2.For the modified second-order discretization, MV = DV makes the total-energy correction unnecessary.Source: §4.2.1 · p. 18
5 Numerical Results
The numerical studies test cG-DeC on a smooth vortex and strong-shock benchmarks, comparing accuracy, efficiency, symmetry, and energy conservation with cG-RK. Results vary with element order and the use of the total-energy correction.
5 Numerical Results
- The evaluation spans Taylor–Green convergence, Sedov accuracy and energy conservation, Noh symmetry, and a three-dimensional triple-point problem.The tests are implemented in Laghos using MFEM.Source: §5 · p. 18
5.1 Taylor-Green Vortex
- cG-DeC is more efficient overall for Q1Q0/Q2Q1; expected convergence requires five Q3Q2 corrections, ten Q4Q3 corrections, and 37 B3B2 corrections.The high B3B2 correction count is consistent with the iteration’s dependence on powers of AV and its spectral radius being near one; one Neumann-series Laplacian iteration sufficed across the tests.Source: §5.1 · pp. 20–21, Figure 2 · p. 20, Table 1 · p. 21
5.2 Sedov
- At fixed kinematic DoFs, cG-DeC matches cG-RK accuracy, is slightly more accurate for Q1Q0/Q2Q1 with M=1, K=2, and has better energy conservation for Q4Q3 with M=3, K=10 versus cG-RK using RK4.With M=1, K=2, applying δF gives good energy conservation but increases error compared with cG-DeC.Source: §5.2 · pp. 21–22, Table 2 · p. 22
- The Sedov strong-shock comparison finds that theoretical correction counts change error little while substantially improving cG-DeC efficiency.For higher orders, the study contrasts theoretical counts with counts needed for smooth-problem convergence.Source: §5.2 · p. 22
5.3 Noh
- Noh results are remarkably similar between methods, with slightly better symmetry from cG-DeC; δF improves Q1Q0 shock location but worsens Q2Q1 symmetry.The test’s reference postshock density is 16 at a shock radius of 0.2 at t = 0.6.Source: §5.3 · pp. 24–25
5.4 Triple Point
- On the triple-point test, cG-DeC robustly resolves the fields on the lower-resolution mesh, while cG-RK results are visually indistinguishable.The comparison uses Q4Q3 on a 16 × 8 × 8 hexahedral mesh.Source: §5.4 · pp. 25–26
6 Conclusion
The conclusion reports comparable accuracy between cG-DeC and cG-RK on several benchmarks, alongside efficiency gains in some settings and robust solutions on the Triple Point problem. It also identifies spectral-radius dependence as relevant to DeC convergence.
6 Conclusion
- On Taylor-Green, cG-DeC is more efficient for Q1Q0 and Q2Q1, while extra corrections reduce efficiency for kinematic polynomial orders greater than two.The extra corrections are measured against the theoretical number found in.Source: §6 · p. 26
- On Sedov, cG-DeC and cG-RK have close to identical L1 errors across polynomial orders, while energy correction increases measured L1 errors.Using the theoretical correction count has little impact on global L1 errors while improving efficiency.Source: §6 · p. 26
- On Noh, the methods capture peak postshock density well and preserve symmetry.Source: §6 · p. 26
- The Q4Q3 Triple Point comparison reports relative L1 differences and total energy differences; cG-DeC provides robust solutions on this complicated problem.Source: Table 5 · p. 26, §6 · pp. 26–27
- DeC convergence in PDEs depends on the spectral radius of A = I − D^-1M, which can approach one depending on polynomial order, dimension, and basis type.The reformulation suggests preconditioning the DeC operators to reduce that spectral radius.Source: §6 · p. 27
Declarations
The authors acknowledge NNSA LDRD support and report no relevant interests or conflicts.
Declarations
- The work acknowledges NNSA LDRD support and declares no relevant financial or non-financial interests or conflicts.Source: Declarations · p. 27