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Stable and Accurate Interpolation Operators for High-Order Multi-Block Finite-Difference Methods

K. Mattsson, Mark H. Carpenter

arXiv:0902.2791v1math.NAcs.NA

TL;DR

The paper addresses how to couple high-order SBP discretizations across blocks when grids or operators differ, while retaining stability, accuracy, and conservation. It develops SBP-preserving interpolation operators and stability conditions for that coupling. Eigenvalue analysis and two-dimensional compressible Euler computations examine the properties, with the computations also identifying careful boundary treatment as necessary for stability.

  • Problem

    Existing multi-block SBP methods constrain interface grids and adjoining operators, motivating coupling that preserves stability, accuracy, and conservation across nonconforming grids or differing schemes.

    Source: §1 · pp. 1–2, §3 · p. 6

  • Method

    The approach combines SBP operators and SAT interface penalties with interpolation operators designed to preserve the SBP property and match the base scheme's accuracy.

    Source: §3 · p. 6, §5 · pp. 17–18

  • Results

    Two-dimensional compressible Euler computations corroborate stability and accuracy, while showing that careful boundary treatment is necessary for stable interface coupling.

    Source: §5 · p. 18

  • Takeaways & Limitations

    The constructed operators are required to preserve the underlying SBP rule and accuracy while using minimal interior width.

    Source: §5 · p. 18

    The extra stability conditions are met for the second- and fourth-order operators, but whether they can be met for sixth- and eighth-order cases remains open.

    Source: §3.1 · p. 12

Abstract

from arXiv · show

Block-to-block interface interpolation operators are constructed for several common high-order finite difference discretizations. In contrast to conventional interpolation operators, these new interpolation operators maintain the strict stability, accuracy and conservation of the base scheme even when nonconforming grids or dissimilar operators are used in adjoining blocks. The stability properties of the new operators are verified using eigenvalue analysis, and the accuracy properties are verified using numerical simulations of the Euler equations in two spatial dimensions.

1 Introduction

High-order finite-difference methods suit long-distance wave propagation, but conventional SBP-SAT multi-block coupling restricts interface grids and operators. This study develops conditions and interpolation operators intended to retain strict stability, accuracy, and conservation while broadening those coupling choices.

1 Introduction

  • Strict stability matters for long-time wave propagation because some convergent, G-K-S stable schemes can still exhibit non-physical solution growth.The introduction defines strict stability as a property of methods that do not allow growth in time.Source: §1 · p. 1
  • The SBP-SAT approach combines SBP finite-difference operators with SAT or projection treatment of physical boundary conditions and extends to multi-block geometries.The stated retained properties are strict stability, accuracy, and conservation.Source: §1 · p. 2
  • Conventional multi-block SBP-SAT methods require matching interface points and identical tangential schemes, restricting nonconforming grids and hybrid discretizations.A prior exception merged a finite-volume scheme by alternating it at the interface.Source: §1 · p. 2
  • The study seeks conditions on adjoining interpolation operators that guarantee interface strict stability, accuracy, and conservation.These conditions form the systematic methodology for coupling SBP methods between blocks.Source: §1 · p. 2
  • The proposed examples couple equal-order SBP operators across 2:1 nonconforming grids and selected unequal-order operators across conforming interfaces.The listed pairs are 2nd:2nd, 4th:4th, 6th:6th, and 8th:8th across nonconforming interfaces, plus 2nd:4th and 4th:8th across conforming interfaces.Source: §1 · p. 3

2 Definitions

This section introduces the one-dimensional SBP framework and its accuracy terminology, then describes how tensor products extend the operators and solution vectors to two-dimensional domains.

2 Definitions

  • The study builds two- and three-dimensional schemes from tensor products of one-dimensional SBP finite-difference operators.The section begins with one-dimensional SBP operators and relevant definitions.Source: §2 · p. 3

2.1 Summation-By-Parts

  • The discrete solution vector stores grid-projected values, and the discrete inner product is defined using a symmetric positive-definite matrix H.The associated squared norm is v^T H v.Source: §2.1 · pp. 3–4
  • A first-derivative SBP operator has the form D1 = H^-1Q, with symmetric positive-definite H and Q + Q^T equal to the boundary matrix B.The boundary matrix has -1 and 1 at the endpoints and zeros elsewhere.Source: §2.1 · p. 4
  • A narrow stencil is an explicit pth-order accurate finite-difference scheme with minimal stencil width; it is narrow-diagonal when its SBP norm matrix H is diagonal.Source: §2.1 · p. 4
  • Interpolation is pth-order accurate when its projected-polynomial error vanishes through degree p−1 in the interior and through degree (p−1)/2 at boundaries.The stated remark says pth-order interpolation preserves the accuracy of narrow-diagonal SBP operators, with numerical experiments consistent with that statement.Source: §2.1 · p. 4

2.2 Two-Dimensional Domains

  • In two dimensions, the solution is organized as vectors along y at each x-grid point, and directional operators are formed using Kronecker products with identity matrices.The section defines separate x- and y-direction operators and norms.Source: §2.2 · pp. 5–6

3 Analysis

The analysis establishes sufficient interface-stability conditions for coupling SBP schemes, then constructs accurate SBP-preserving interpolation operators for common orders and mixed-order pairs. The construction also identifies a higher-order condition that remains unresolved.

3 Analysis

  • Interface stability follows when the penalty matrices satisfy 2ΣL = A and 2ΣR = −A and the interpolation operators satisfy condition (11), assuming correctly implemented outer-boundary terms.The same conditions also fulfill the stated conservation conditions.Source: §3 · pp. 7–8
  • Characteristic coupling is stable with ΣL = A− and ΣR = −A+ when conditions (11) and (12) hold and outer-boundary terms are correctly implemented.The characteristic penalty choice uses the eigendecomposition of the symmetric matrix A.Source: §3 · p. 8
  • Choosing an even α in the modified energy estimate removes the extra conditions (12): stability then requires symmetric X, which follows from condition (11).The resulting coupling is stable for symmetric positive-definite Ω, and SBP-preserving artificial dissipation does not alter the stated stability conditions.Source: §3 · pp. 9–10

3.1 Construction of Interpolation Operators

  • The construction procedure imposes interpolation accuracy and tunes any remaining free parameters to optimize the L2 error.For narrow-diagonal SBP operators, a pth-order interpolation operator preserves the scheme's stated accuracy; other boundary closures may require more accurate interpolation closures.Source: §3.1 · pp. 11–12
  • SBP-preserving operators were constructed for second-, fourth-, sixth-, and eighth-order cases, plus fourth-to-second- and eighth-to-fourth-order couplings.The study considers a 2:1 coarse-to-fine interface ratio; its analysis is not restricted to that ratio.Source: §3.1 · pp. 10, 12
  • Whether the extra stability conditions (12) can be met for sixth- and eighth-order cases remains open; constructed second- and fourth-order operators satisfy them.The paper notes this is the first demonstration, to the authors' knowledge, of stable multidimensional multi-block coupling between finite-difference schemes of different orders.Source: §3.1 · p. 12

4 Computations

Eigenvalue tests distinguish SBP-preserving from non-SBP interpolation, and Euler-vortex simulations report grid-refinement errors and compare mixed-order couplings.

4.1 Numerical Validation of Lemma 3.2

  • Non-SBP interpolation produces eigenvalues right of the imaginary axis, while SBP-preserving interpolation is stable in fourth- and sixth-order tests.The spectra compare the two interpolation types for both discretization orders.Source: Figure 2 · p. 13, §4.1 · p. 13, Figure 3 · p. 14

4.2 Vortex in Free Space

  • A 2-D Euler vortex crosses a two-block nonmatching interface, with coarse-to-fine errors and convergence rates reported for second- through eighth-order schemes.The simulation uses the analytic Euler-vortex solution and advances it to t = 1.Source: §4.2 · p. 14, Table 1 · p. 16, Table 2 · p. 16
  • The eighth/fourth-order hybrid results are almost identical to eighth order in both blocks; in the fourth/second versus fourth-order comparison, second-order error clearly dominates.These comparisons are reported for Tables 6 versus 4 and Tables 5 versus 2, respectively.Source: Table 4 · p. 16, Table 2 · p. 16, §4.2 · p. 17

5 Conclusions

The paper uses SBP, SAT, and SBP-preserving interpolation to couple blocks on nonconforming grids while targeting stability, accuracy, and simplicity. Its two-dimensional Euler computations support the coupling’s stability and accuracy, while showing that boundary treatment matters for stability.

5 Conclusions

  • The approach combines SBP operators, SAT, and SBP-preserving interpolation to enforce interface conditions for general hyperbolic and parabolic problems on nonconforming grids.Source: §5 · p. 17
  • The operators are designed for stability by construction without changing existing SBP schemes, while maintaining convergence and simplicity.Source: §5 · p. 17
  • The 2-D compressible Euler computations corroborate stability and accuracy, but careful boundary treatment is necessary for stable interface coupling.The interpolation operators preserve the SBP rule, match the underlying scheme’s accuracy order, and use minimal-width interiors.Source: §5 · p. 18

I Numerical Validation of Lemma 3.3 and Lemma 3.4

The sixth-order characteristic coupling shows positive-real-part eigenvalues, whereas quadratic coupling yields non-positive-real-part eigenvalues. The comparison tests the extra stability conditions and the modified penalties used to avoid them.

Numerical Validation of Lemma 3.3 and Lemma 3.4

  • Quadratic penalties (14) remove the need for condition (12), with stability following when Ω is symmetric positive definite; the study uses Ω = Ā.Source: I Numerical Validation of Lemma 3.3… · p. 19
  • The largest positive real part for sixth-order characteristic coupling is of order 10^-4, while quadratic coupling has only non-positive-real-part eigenvalues.The reported eigenvalue spectrum uses 212 unknowns in the left coarse block and scales eigenvalues by the fine-grid spacing hR.Source: I Numerical Validation of Lemma 3.3… · p. 19, Figure 5 · p. 20
  • The sixth-order characteristic coupling’s positive-real-part eigenvalues indicate that condition (12) is necessary for that coupling.Second- and fourth-order characteristic cases obey condition (12) and are reported to have no positive-real-part eigenvalues.Source: I Numerical Validation of Lemma 3.3… · p. 19

II Parabolic problems

The paper extends its discussion to parabolic systems with second-derivative terms and continuity conditions at the interface. It introduces a corresponding semi-discrete approximation.

Parabolic problems

RDx

The RDy discussion states stability conditions for the viscous multi-block scheme, and the subsequent cases present interpolation operators across several orders. The sixth-order case notes that the upwind stability condition could not be achieved.

RDy

  • The viscous multi-block scheme is stable when the inviscid stability lemma conditions hold, the outer boundary terms are correctly implemented, and Σ2 and Σ4 are symmetric.Source: RDy · p. 21

Second-Order case

Sixth-Order Case

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