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ADER-ZIST - a mixed explicit ADER-DG and zonally implicit space-time DG approach for Cartesian cut cell meshes

Michael Dumbser, Christian Engwer, Sandra May, Peter Munch

arXiv:2610.13245v1math.NA

TL;DR

Explicit CFL limits on Cartesian cut-cell meshes shrink with the smallest cells, restricting the time step. ADER-ZIST couples explicit ADER-DG away from embedded boundaries to implicit space-time DG on cut cells and their direct neighbors. The reported tests attain nominal order N + 1 in one-dimensional cut-cell meshes.

  • Problem

    Arbitrarily small cut cells make explicit CFL time steps shrink with the smallest cell, making a background-mesh time step unstable.

    Source: §1 · p. 2, §2.1 · p. 5

  • Method

    ADER-ZIST uses explicit ADER-DG away from embedded boundaries and a zonally implicit space-time DG scheme on cut cells and their direct neighbors, coupled through numerical fluxes.

    Source: §1 · p. 3, §2.3.1 · p. 9

  • Results

    N + 1 nominal order is attained on the tested one-dimensional cut-cell meshes, with errors comparable to unmixed schemes on equidistant meshes.

    Source: §4 · p. 38

  • Takeaways & Limitations

    The conservative coupling allows a time step determined by background-cell size independently of cut-cell size in the reported experiments.

    Source: §1 · p. 3

    Extensions to three dimensions, moving embedded boundaries, and more general hyperbolic systems remain future work.

    Source: §4 · p. 38

Abstract

from arXiv · show

We present high order accurate mixed explicit-implicit discontinuous Galerkin (DG) schemes for the solution of hyperbolic systems of conservation laws on Cartesian cut cell meshes. Such meshes may contain arbitrarily small cut cells, which leads to the well-known small cell problem: the CFL condition of explicit schemes would severely restrict the admissible time step. To overcome this difficulty, we use high order explicit fully-discrete one-step ADER-DG schemes only on Cartesian cells away from the embedded boundaries. The zone close to the embedded boundary, which consists of cut cells and their direct face neighbors, is discretized with an implicit space-time DG scheme that is coupled to the explicit scheme directly via the numerical flux. The resulting scheme, called ADER-ZIST (ADER-DG with Zonally Implicit Space-Time DG), is conservative by construction, and its time step is determined only by the size of the Cartesian background cells. Curved embedded boundaries are represented by high order isoparametric sub-triangles inside the cut cells, and the nonlinear system of the implicit cells is solved via an inexact matrix-free Newton-Krylov method. For problems with shock waves, we add an artificial viscosity term whose strength is controlled by an oscillation-eliminating smoothness sensor. ADER-ZIST is assessed on a comprehensive set of test problems for the compressible Euler equations, ranging from smooth flows around straight and curved embedded boundaries to acoustic scattering and shock-dominated flows. The scheme essentially reaches its nominal order of accuracy N+1 for smooth solutions and for polynomial degrees up to N=5, on meshes with minimal volume fractions of the small cut cells down to the order of 10^-7.

1. Introduction

Arbitrarily small cut cells make explicit CFL time steps impractically restrictive, while existing remedies trade mesh simplicity, accuracy, or local time stepping. ADER-ZIST instead couples explicit ADER-DG with implicit space-time DG near embedded boundaries, preserving a common background-mesh time step.

1. Introduction

  • Explicit CFL limits shrink with the smallest cut cell, making a background-mesh time step unstable there.Cell merging can address the issue but shifts complexity to mesh generation, particularly in three dimensions.Source: §1 · p. 2
  • Earlier DG remedies include implicit Euler, ghost penalties, Domain of Dependence stabilization, state redistribution, and Galerkin difference methods.The cited approaches target stability or accuracy through distinct mechanisms, including penalties and extended basis functions.Source: §1 · p. 2
  • ADER-DG uses an element-local space-time predictor followed by a fully discrete one-step corrector that couples neighboring cells through numerical fluxes.Its space-time structure also supports high-order local time stepping.Source: §1 · pp. 2–3
  • ADER-ZIST treats cut cells and their direct neighbors implicitly and the remaining Cartesian cells explicitly, coupling the schemes through numerical fluxes.The resulting fully discrete one-step method is conservative, with a time step determined by background cells rather than cut-cell size.Source: §1 · p. 3
  • For shock problems, the method adds artificial viscosity controlled by an oscillation-eliminating smoothness sensor.The paper also describes curved-boundary treatment and implementation details for the mixed scheme.Source: §1 · p. 3

2. Mixed explicit ADER-DG and zonally implicit space-time DG approach

The section presents the ADER-ZIST design, combining explicit ADER-DG on regular cells with zonally implicit space-time DG near embedded boundaries. It also covers shock capturing and implementation, while restricting the study to two dimensions.

2. Mixed explicit ADER-DG and zonally implicit space-time DG approach

  • The method combines explicit ADER-DG away from embedded boundaries with zonally implicit space-time DG on cut cells and their neighbors.Both schemes use the same space-time weak formulation and local polynomial degree N; they differ in when neighbor interactions enter.Source: §2 · p. 4
  • The section also presents artificial viscosity for shock waves and implementation aspects including curved boundaries and solving the implicit nonlinear system.Conditioning of that nonlinear system is also discussed.Source: §2 · p. 4
  • The paper develops the approach for a two-dimensional setting, although its basic ideas may apply in three dimensions.The section introduces the equations, mesh, discrete spaces, numerical flux, and conservation property.Source: §2 · p. 4

2.1. Governing equations, cut cell mesh and notation

The section sets up the two-dimensional compressible Euler problem on a Cartesian background mesh intersected by an embedded domain. It defines the cut-cell geometry, approximation spaces, and numerical flux used to couple explicit and implicit cells.

2.1. Governing equations, cut cell mesh and notation

  • The governing system is the two-dimensional compressible Euler equations, with conserved variables and fluxes defined for admissible states.The ideal-gas equation of state uses the ratio of specific heats γ.Source: §2.1 · p. 4
  • The mesh intersects a Cartesian background grid with the domain; cut cells are polygonal, may be non-convex, and can have arbitrarily small volume fractions.Cells with αE < 10^-9 are removed, but otherwise small cells are neither merged nor assigned a lower polynomial degree.Source: §2.1 · pp. 4–5
  • The explicit CFL step scales with the smallest cell, so tiny cut cells make a background-mesh step unstable without a special treatment.ADER-ZIST uses explicit ADER-DG on regular cells and an implicit space-time DG scheme on cut cells and their direct neighbors, with a time step independent of cut-cell size in the reported experiments.Source: §2.1 · pp. 5–6
  • Space-time polynomials of degree N are used on both Cartesian and cut cells, with tensor-product nodal bases on Cartesian cells and modal total-degree bases on cut cells.The restriction of each space-time polynomial to the end of a time slab defines the solution at the new time level.Source: §2.1 · p. 6
  • A four-argument Rusanov flux couples neighboring cells and is designed to be consistent and conservative when both sides use the same frozen signal-speed estimate.The speed is computed from old-time traces and remains fixed through the implicit Newton iterations.Source: §2.1 · pp. 6–7

2.2. A common starting point: weak formulation of the PDE system in space-time

Both schemes begin from a space-time weak formulation, but differ in whether neighboring-cell interactions are included within the time slab or in a later explicit corrector. The local ADER-DG predictor supports explicit updates away from cut cells, while space-time DG couples cells implicitly near the boundary.

2.2. A common starting point: weak formulation of the PDE system in space-time

  • Both methods multiply the governing PDE by a space-time test function and integrate over each space-time cell, using the known solution at the slab's initial time.The distinction is how neighboring cells enter: within-slab coupling gives an implicit system, while a later corrector gives an element-local explicit update.Source: §2.2 · p. 7

2.2.1. Explicit scheme: ADER-DG

  • ADER-DG forms an element-local space-time predictor by integrating the weak formulation by parts in time while leaving the flux divergence in strong form.The predictor uses only information inside its cell and approximates the local Cauchy problem.Source: §2.2.1 · pp. 7–8
  • The predictor's nonlinear system is solved by Picard iteration, with N + 2 iterations used in the implementation to retain nominal accuracy in the reported experiments.For linear homogeneous constant-coefficient systems, the iteration is exact after finitely many steps; for nonlinear systems, convergence requires a sufficiently small time step.Source: §2.2.1 · p. 8
  • The ADER corrector uses known predictors from the cell and its neighbors, yielding an explicit fully discrete one-step update of nominal order N + 1.On a small cut cell, the explicit CFL condition and predictor-iteration convergence can both fail at the background-mesh step.Source: §2.2.1 · p. 8

2.2.2. Implicit scheme: space-time DG

  • Space-time DG includes neighboring-cell unknowns through numerical fluxes within the time slab, producing a coupled nonlinear system on the implicit region.Cartesian transition cells use the stated space-time formulation, while cut cells use its strong form.Source: §2.2.2 · pp. 8–9
  • For cut cells, the strong form uses the projected flux in the boundary term; subtracting the unprojected flux could destroy conservation for nonlinear fluxes.The text identifies the componentwise L2 projection as the projection used in this formulation.Source: §2.2.2 · p. 9
  • The implicit space-time DG scheme is not subject to the explicit cut-cell CFL condition, and the reported computations use a background-mesh time step, with an added restriction when artificial viscosity is active.The implicit face integrals involve neighboring cells' unknown space-time polynomials.Source: §2.2.2 · p. 9

2.3. The mixed explicit-implicit ADER-space-time DG scheme

ADER-ZIST treats cut cells and their direct face neighbors implicitly while using explicit ADER-DG elsewhere, coupling the two schemes through numerical fluxes. Its three-step update combines local explicit predictors with a zonal implicit solve and is conservative under stated assumptions.

2.3.1. Main idea and classification of cells

  • Cut cells and their face-neighbor transition cells use implicit space-time DG, while all remaining cells use explicit ADER-DG, coupled through numerical fluxes.No cut cell shares a face with an explicit cell.Source: §2.3.1 · p. 9

2.3.2. The three-step algorithm

  • The update computes local explicit predictors, solves one nonlinear system for the implicit region, then corrects explicit cells using implicit-region traces.The implicit system covers only the cut cells and their direct neighbors; in two dimensions, their count grows like O(h^-1), compared with O(h^-2) for the mesh.Source: §2.3.2 · pp. 10–11

2.3.3. Conservation

  • With matching face quadrature and exact polynomial integration and implicit solves, the mixed scheme is conservative; in computations, curved-cell quadrature and solver residuals produce small balance defects.The reported relative mass and total-energy balance defect is below 2 · 10^-11 per time step, including on curved cut cells.Source: §2.3.3 · pp. 11–12

2.3.4. Why ADER-DG and space-time DG are a good match for explicit-implicit coupling

  • ADER predictors and implicit space-time DG polynomials share a space-time representation, so their interface fluxes couple directly without temporal interpolation.The ADER predictor corresponds to the space-time DG strong form with spatial-face fluctuations switched off.Source: §2.3.4 · p. 13

2.4. Shock capturing via artificial viscosity

For shock problems, ADER-ZIST adds artificial viscosity controlled by a smoothness sensor. The sensor responds to jumps across cell faces, while local directional scaling accommodates cut-cell geometry and keeps the added viscosity vanishing under refinement.

2.4. Shock capturing via artificial viscosity

  • For shock problems, artificial viscosity regularizes the equations instead of using a subcell limiter that is cumbersome on cut cells and in the coupled implicit region.The viscosity is switched on only for the shock-wave test problems.Source: §2.4 · p. 14
  • The viscous terms use the non-symmetric interior penalty Galerkin method, with viscosity contributions added to implicit equations and explicit correctors.On explicit cells, the local predictor receives only the volume contribution, without face terms.Source: §2.4 · p. 14
  • Viscosity is scaled separately by each cell's local x- and y-extent, accommodating cut cells with strongly different directional dimensions while vanishing under mesh refinement.The coefficients are constant per cell and time step and are applied componentwise to conserved variables.Source: §2.4 · pp. 14–15
  • The added parabolic term imposes an additional time-step restriction on explicit cells, but that restriction depends only on background-cell sizes.The explicit-cell condition is added to the existing CFL restriction.Source: §2.4 · p. 15
  • The oscillation-eliminating sensor measures solution and derivative jumps across cell faces; it indicates smooth regions near one and approaches zero near discontinuities.The raw sensor is spread to neighboring cells to form the viscosity-control sensor.Source: §2.4 · p. 16

2.5. Implementation aspects

The implementation combines curved cut-cell geometry and compatible quadrature with a zonally implicit solve. The implicit system uses a matrix-free Newton–Krylov method, while bounding-box scaling and block Jacobi preconditioning address conditioning concerns.

2.5.1. Cut cell geometry and quadrature

  • Curved boundaries are represented by cubic splines and isoparametric subtriangles, with degree-dependent quadrature for volume and face integrals.The boundary points are assumed sufficiently dense for the cubic spline approximation to be accurate.Source: §2.5.1 · p. 16

2.5.2. Basis functions

  • Cartesian and transition cells use nodal tensor-product Lagrange bases, while cut cells use modal space-time polynomials of degree up to N.The time quadrature points are shared so explicit–implicit face fluxes are evaluated at matching temporal nodes.Source: §2.5.2 · p. 16, §2.5.1 · p. 16

2.5.3. Solving the implicit system

  • The transition and cut-cell degrees of freedom are solved together using an inexact Newton method with matrix-free Jacobian–vector products and preconditioned GMRES.A block Jacobi preconditioner uses one block per implicit cell; its blocks are exactly inverted and periodically recomputed.Source: §2.5.3 · p. 17

2.5.4. Conditioning

  • Conditioning is addressed through bounding-box scaling and block Jacobi preconditioning, but a detailed analysis of the complete discrete operator is left to future work.Cells with volume fraction below 10^-9 are removed; otherwise, polynomial degree is not reduced on small cut cells.Source: §2.5.4 · pp. 17–18

3. Numerical Results

The numerical section benchmarks ADER-ZIST on accuracy, robustness, and performance, using smooth-flow, acoustic-scattering, and shock problems. The setup keeps the time-step restrictions dependent on background-cell size rather than cut-cell size.

3. Numerical Results

  • The benchmark suite covers one-dimensional accuracy, smooth two-dimensional flows with straight and curved boundaries, acoustic scattering among five cylinders, and shock-dominated flows.Computational costs are also compared with globally implicit space-time DG and, for linear advection, explicit Runge–Kutta DG with DoD stabilization.Source: §3 · p. 18

3.1. Setup

  • The time-step restrictions depend only on background-cell size because cut cells and their direct neighbors are treated implicitly while other cells use explicit ADER-DG.The stated default setup uses the compressible Euler equations with γ = 1.4 and CFL = 0.4.Source: §3.1 · p. 18
  • The one-dimensional meshes place one small cut cell in each block of 20 background cells, doubling the number of blocks under refinement.All results use two independently written Fortran codes, with an independent deal.II implementation used to cross-check results.Source: §3.1 · p. 18, Figure 5 · p. 19

3.2. Smooth problem in one dimension

In one-dimensional advection tests, ADER-ZIST retains nominal order N + 1 despite using cut cells and an explicit–implicit transition. The transition changes the leading local error, but the tested coupling does not reduce the one-step convergence order.

3.2.1. Convergence study

  • ADER-ZIST achieves convergence consistent with nominal order N + 1 in L1 and L∞, with errors comparable to both unmixed equidistant-mesh schemes.The result is reported after one step and at final time; L∞ orders are less regular because the maximum error occurs in different cell types across meshes.Source: §3.2.1 · p. 19, Table 1 · p. 20

3.2.2. Coupling of the explicit and the implicit scheme

  • The coupling test isolates its local effect by comparing one-step updates from identical states in the first explicit cell downstream of the implicit region.The setup uses three implicit cells and an equidistant mesh without a small cell, with N = 1, 2, 3.Source: §3.2.2 · pp. 19, 21
  • The coupling contribution D and both schemes’ local errors converge with order N + 1, although the coupling changes the leading local error and can increase or decrease it.For N = 1, 2, 3, D/h^(N+1) approaches approximately 0.41, 0.28, and 0.14, respectively; D/eADER approaches 0.39, 0.48, and 0.22.Source: §3.2.2 · p. 21, Table 2 · p. 22
  • ADER-ZIST’s shared space-time weak formulation lets the explicit and implicit schemes differ in the flux polynomial without enlarging the implicit region for an element-local ADER predictor.The related MPRKC–Trap method likewise avoids reduced one-step order at the transition, but its two-stage explicit scheme required a larger implicit region.Source: §3.2.2 · p. 21

3.3. Smooth problems in two dimensions

The section tests ADER-ZIST on a straight ramp and then on curved and non-convex boundaries, using smooth-flow cases to examine its accuracy.

3.3.1. Advection of a smooth density profile along a ramp

  • Ramp-test observed orders between successive meshes are irregular, especially in L∞, as minimum volume fractions range from 7.76 · 10^-8 to 5.54 · 10^-3.Least-squares fits use a denser mesh sequence; the number of time steps grows linearly with Imax and is not affected by αmin.Source: §3.3.1 · pp. 22–23

3.3.2. Isentropic vortex around a cylinder

  • For the isentropic vortex around a cylinder, L1 rates are close to N + 1, while L2 and L∞ errors behave more irregularly.The minimum volume fraction ranges from 1.07 · 10^-3 to 2.55 · 10^-1.Source: §3.3.2 · p. 24

3.3.3. Ringleb flow

  • In the non-convex Ringleb channel, observed rates are close to N + 1 in all norms except L∞ for N = 5.The authors report that the results confirm accurate curved-boundary representation and the method's ability to handle this challenging domain.Source: §3.3.3 · p. 26

3.3.4. Supersonic vortex

  • For N = 3, the reported rates lie between 3.88 and 4.51, reaching nominal order N + 1, the expected order for that degree.For N = 2, rates lie between 2.52 and 2.62, below nominal but slowly increasing under refinement. For N = 4 and N = 5, L2 least-squares fits give about 5.51 and 6.16, respectively; L∞ errors are more irregular.Source: §3.3.4 · p. 28

3.3.5. Low Mach number flow around a cylinder

  • For the low-Mach cylinder, ρ, u, and v achieve order 3 on all meshes, while pressure does so only on the first refinement.On finer meshes, pressure discretization error reaches the difference between the incompressible reference and the compressible Euler solution.Source: §3.3.5 · p. 30

3.4. Acoustic wave propagation in a more complex geometry: pressure pulse scattered by five cylinders

The five-cylinder test examines small-amplitude acoustic scattering in a non-convex domain with multiple obstacles. ADER-ZIST reproduces the reference pressure profile closely along one obstacle.

3.4. Acoustic wave propagation in a more complex geometry: pressure pulse scattered by five cylinders

  • The five circular obstacles lie on a circle around the domain center, and the small-amplitude Gaussian pressure pulse is simulated to T = 6.Each cylinder has radius R = 2; the pressure and density perturbation amplitude is A = 10^-4.Source: §3.4 · p. 31
  • The pressure-deviation comparison agrees very well: RMS difference is below 1% of the reference maximum absolute scaled deviation, and local-extrema amplitudes agree within 2%.The comparison uses ADER-ZIST with N = 3 on 106 × 106 cells and the DG-P5Q5 solution on 53 × 53 cells.Source: Figure 13 · p. 32, §3.4 · p. 32
  • The solution shows the expected five-fold symmetric scattering pattern, with no visible spurious oscillations or asymmetries near embedded boundaries.The qualitative pattern agrees with the results in.Source: §3.4 · p. 32

3.5. Test problems with shocks

Shock tests show consistent solutions across the Cartesian, transition, and cut cells, and good agreement with reference density profiles for shock diffraction around a cylinder. Artificial viscosity is activated near shocks and contact discontinuities.

3.5.1. Sod shock tube with walls

  • The Sod test uses a 100 × 10 background mesh with walls producing cut cells of volume fraction α = 0.1, and employs N = 2 through T = 0.4.Explicit Cartesian, implicit transition, and implicit cut cells use different bases or schemes in this test.Source: §3.5.1 · pp. 32–33
  • Cut-cell and transition-cell values agree with Cartesian-cell values and the exact solution; artificial viscosity activates only near the shock and contact discontinuity.Source: §3.5.1 · p. 33

3.5.2. Shock diffraction around a cylinder

  • The cylinder test models a right-moving Mach 2 shock diffracting around a cylinder, using N = 2 on a 150 × 165 mesh through T1 = 0.25 and T2 = 0.3.Source: §3.5.2 · p. 33
  • Mean absolute density differences were 0.029 (0.6% of reference range) at T1 = 0.25 and 0.043 (0.8%) at T2 = 0.3 versus the two reference solutions.The central density spike at t = 0.3 is lower than in the reference, probably due to first-order artificial viscosity versus the reference's second-order FV scheme.Source: §3.5.2 · p. 34

3.6. Computational cost

The cost comparisons find that ADER-ZIST produces similar solutions to globally implicit space-time DG at lower computational cost, with savings increasing under mesh refinement. Against DoD stabilization, it is more expensive on linear advection but applies to nonlinear multidimensional systems.

3.6. Computational cost

  • Global implicit runs cost 1.0–5.2 times as much as ADER-ZIST, while relative L1 differences are 4.8·10^-3 for pressure perturbation and 1.9·10^-3 for density.The comparison uses the same meshes, time step, and number of time steps; Table 8 reports the computational-cost comparison.Source: §3.6 · p. 35, Table 8 · p. 37
  • Implicit cells cost about 12–30 times as much per cell and time step as explicit cells, and account for 70% of scattering CPU time despite being 8% of cells.The reported implicit-cell share of total CPU time is 88%–98% for the Ringleb flow.Source: §3.6 · p. 36
  • On linear advection, ADER-ZIST's L1 errors are 3%–18% smaller than DoD's, but its CPU time is 1.4–2.7 times higher.All three schemes converge at nominal order N + 1; ADER-ZIST also applies to nonlinear multidimensional systems, unlike the stated development scope of DoD.Source: §3.6 · p. 36, Table 9 · p. 37

4. Conclusion

ADER-ZIST treats boundary-adjacent cut cells implicitly and the remaining Cartesian cells explicitly, addressing the small-cell problem while retaining a background-mesh time step. Tests show nominal-order accuracy in most cases and speedups over globally implicit space-time DG, while several extensions and theoretical analyses remain future work.

4. Conclusion

  • ADER-ZIST couples explicit ADER-DG on regular cells with implicit space-time DG on cut cells and their Cartesian face neighbors, conserving fluxes with a time step set by background-cell size.The implicit system is solved by an inexact matrix-free Newton–Krylov method; curved boundaries use isoparametric sub-triangles, and shock cases add sensor-controlled artificial viscosity.Source: §4 · p. 38
  • Tests reach nominal order N + 1 for degrees up to N = 5 in most cases, including cut cells with volume fractions down to about 10^-7; speedup over globally implicit DG reaches 5.2.For linear advection, ADER-ZIST costs 1.4–2.7 times as much as DoD stabilization, with comparable L1 errors, while applying to nonlinear multidimensional systems.Source: §4 · p. 38
  • Three-dimensional and moving-boundary extensions, more general systems including non-conservative ones, space-time AMR, and theoretical stability and accuracy analyses remain future work.Source: §4 · p. 38

Declaration of generative AI and AI-assisted technologies

The authors report AI assistance with drafting, literature work, language, and code, and say they reviewed the material and take responsibility for the article and numerical results.

Declaration of generative AI and AI-assisted technologies

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