Pieter D. Boom

A generalized framework for nodal first derivative summation-by-parts operators

A generalized framework is presented that extends the classical theory of finite-difference summation-by-parts (SBP) operators to include a wide range of operators, where the main extensions are (i) non-repeating interior point operators, (ii) nonuniform nodal distribution in the computational domain, (iii) operators that do not include one or both boundary nodes. Necessary and sufficient conditions are proven for the existence of nodal approximations to the first derivative with the SBP property. It is proven that the positive-definite norm matrix of each SBP operator must be associated with a quadrature rule; moreover, given a quadrature rule there exists a corresponding SBP operator, where for diagonal-norm SBP operators the weights of the quadrature rule must be positive. The generalized framework gives a straightforward means of posing many known approximations to the first derivative as SBP operators; several are surveyed, such as discontinuous Galerkin discretizations based on the Legendre-Gauss quadrature points, and shown to be SBP operators. Moreover, the new framework provides a method for constructing SBP operators by starting from quadrature rules; this is illustrated by constructing novel SBP operators from known quadrature rules. To demonstrate the utility of the generalization, the Legendre-Gauss and Legendre-Gauss-Radau quadrature points are used to construct SBP operators that do not include one or both boundary nodes.

HIGH-ORDER IMPLICIT TIME-MARCHING METHODS BASED ON GENERALIZED SUMMATION-BY-PARTS OPERATORS ∗

This article extends the theory of classical finite-difference summation-by-parts (FD-SBP) time-marching methods to the generalized summation-by-parts (GSBP) framework. Dual-consistent GSBP time-marching methods are shown to retain: A and L-stability, as well as superconvergence of integral functionals when integrated with the quadrature associated with the discretization. This also implies that the solution approximated at the end of each time step is superconvergent. In addition GSBP time-marching methods constructed with a diagonal norm are BN-stable. This article also formalizes the connection between FD-SBP/GSBP time-marching methods and implicit Runge-Kutta methods. Through this connection, the minimum accuracy of the solution approximated at the end of a time step is extended for nonlinear problems. It is also exploited to derive conditions under which nonlinearly stable GSBP time-marching methods can be constructed. The GSBP approach to time marching can simplify the construction of high-order fully-implicit Runge-Kutta methods with a particular set of properties favourable for stiff initial value problems, such as L-stability. It can facilitate the analysis of fully discrete approximations to PDEs and is amenable to to multi-dimensional spcae-time discretizations, in which case the explicit connection to Runge-Kutta methods is often lost. A few examples of known and novel Runge-Kutta methods associated with GSBP operators are presented. The novel methods, all of which are L-stable and BN-stable, include a four-stage seventh-order fully-implicit method, a three-stage third-order diagonally-implicit method, and a fourth-order four-stage diagonally-implicit method. The relative efficiency of the schemes is investigated and compared with a few popular non-GSBP Runge-Kutta methods.

Time-Accurate Flow Simulations Using an Efficient Newton-Krylov-Schur Approach with High-Order Temporal and Spatial Discretization

In order to demonstrate the potential advantages of high-order spatial and temporal discretizations, implicit large-eddy simulations of the Taylor-Green vortex ow and transi- tional ow over an SD7003 wing are computed using a variable-ordernite-difference code on multi-block structured meshes. The spatial operators satisfy the summation-by-parts property, with block interface coupling and boundary conditions enforced through simul- taneous approximation terms. The solution is integrated in time with explicit-�rst-stage, singly-diagonally-implicit Runge-Kutta methods. Simulations of the Taylor-Green vortex show the clear advantage of high-order spatial discretizations in terms of accuracy and effi- ciency. The higher-order methods are better able to delay excessive dissipation on coarser grids and are better able to capture the details of the ow onner grids. Similar dissipa- tion and enstrophy proles are obtained with a second-order spatial discretization, and a fourth-order spatial discretization with half the number of grid points in each direction, half the number of time steps, and approximately 85% less CPU time. Temporal convergence studies demonstrate the relatively high efficiency of the fourth-order explicit-�rst-stage, singly-diagonally-implicit Runge-Kutta method, except for simulations requiring only a minimum level of accuracy. Results of the simulation of transitional ow over the SD7003 wing show good agreement with experiment and other computations, despite a relatively coarse grid. The use of high-order discretizations is shown to be essential in obtaining this accuracy efficiently. These results give a clear picture of the benets of high-order discretizations, along with the advantages of the novel parallel Newton-Krylov-Schur algo- rithm presented, for high-accuracy unsteady ow simulation.

Results from the Fifth AIAA Drag Prediction Workshop obtained with a parallel Newton-Krylov-Schur flow solver discretized using summation-by-parts operators

We present the solution of the test cases from the Fifth AIAA Drag Prediction Workshop computed with a novel Newton-Krylov-Schur parallel solution algorithm for the ReynoldsAveraged Navier-Stokes equations coupled with the Spalart-Allmaras one-equation turbulence model. The algorithm employs summation-by-parts operators on multi-block structured grids, while simultaneous approximation terms are used to enforce boundary conditions and coupling at block interfaces. Two-dimensional verification and validation cases highlight the correspondence of the current algorithm to established flow solvers as well as experimental data. The common grid study, using grids with up to 150 million nodes around the NASA Common Research Model wing-body configuration, demonstrates the parallel computation capabilities of the current algorithm, while the buffet study demonstrates the ability of the solver to compute flow with substantial recirculation regions and flow separation. The use of quadratic constitutive relations to modify the Boussinesq approximation is shown to have a significant impact on the recirculation patterns observed at higher angles of attack. The algorithm is capable of efficiently and accurately calculating complex three-dimensional flows over a range of flow conditions, with results consistent with those of well-established flow solvers using the same turbulence model.

High-Order Implicit Temporal Integration for Unsteady Compressible Fluid Flow Simulation

This paper presents an overview of high-order implicit time integration methods and their associated properties with a specific focus on their application to computational fluid dynamics. A framework is constructed for the development and optimization of general implicit time integration methods, specifically including linear multistep, Runge-Kutta, and multistep Runge-Kutta methods. The analysis and optimization capabilities of the framework are verified by rederiving methods with known coefficients. The framework is then applied to the derivation of novel singly-diagonally-implicit Runge-Kutta methods, explicit-first-stage singly-diagonally implicit Runge-Kutta methods, and singly-diagonallyimplicit multistep Runge-Kutta methods. The fourth-order methods developed have similar efficiency to contemporary methods; however a fifth-order explicit-first-stage singlydiagonally-implicit Runge-Kutta method is obtained with higher relative efficiency. This is confirmed with simulations of van der Pol’s equation.

Corner-corrected diagonal-norm summation-by-parts operators for the first derivative with increased order of accuracy

Combined with simultaneous approximation terms, summation-by-parts (SBP) operators offer a versatile and efficient methodology that leads to consistent, conservative, and provably stable discretizations. However, diagonal-norm operators with a repeating interior-point operator that have thus far been constructed suffer from a loss of accuracy. While on the interior, these operators are of degree 2p, at a number of nodes near the boundaries, they are of degree p, and therefore of global degree p - meaning the highest degree monomial for which the operators are exact at all nodes. This implies that for hyperbolic problems and operators of degree greater than unity they lead to solutions with a global order of accuracy lower than the degree of the interior-point operator. In this paper, we develop a procedure to construct diagonal-norm first-derivative SBP operators that are of degree 2p at all nodes and therefore can lead to solutions of hyperbolic problems of order 2 p + 1 . This is accomplished by adding nonzero entries in the upper-right and lower-left corners of SBP operator matrices with a repeating interior-point operator. This modification necessitates treating these new operators as elements, where mesh refinement is accomplished by increasing the number of elements in the mesh rather than increasing the number of nodes. The significant improvements in accuracy of this new family, for the same repeating interior-point operator, are demonstrated in the context of the linear convection equation.

Optimization of high-order diagonally-implicit Runge–Kutta methods

Abstract This article presents constrained numerical optimization of high-order linearly and algebraically stable diagonally-implicit Runge–Kutta methods. After satisfying the desired order conditions, undetermined coefficients are optimized with respect to objective functions which consider accuracy, stability, and computational cost. Constraints are applied during the optimization to enforce stability properties, to ensure a well-conditioned method, and to limit the domain of the abscissa. Two promising third-order methods are derived using this approach, labelled SDIRK[3,(1,2,2)](3)L_14 and SDIRK[3,1](4)L_SA_5. Both optimized schemes have a good balance of properties. The relative error norm of the latter, the L 2 -norm scaled by a function of the number of implicit stages, is a factor of two smaller than comparable methods found in the literature. Variations on these methods are discussed relative to trade-offs in their accuracy and stability properties. A novel fifth-order scheme SDIRK[5,1](5)L_02 is derived with a significantly lower relative error norm than the comparable fifth-order A-stable reference method. In addition, the optimized scheme is L-stable. The accuracy and relative efficiency of the Runge–Kutta methods are verified through numerical simulation of van der Pols equation, as well as numerical simulation of vortex shedding in the laminar wake of a circular cylinder, and in the turbulent wake of a NACA 0012 airfoil. These results demonstrate the value of numerical optimization for selecting undetermined coefficients in the construction of high-order Runge–Kutta methods with a balance between competing objectives.

Tensor-Product Summation-by-Parts Operators

This paper presents a numerical investigation of the tradeo↵s between various discretization approaches and operators, based on diagonal-norm summation-by-parts (SBP) operators, using the two-dimensional linear convection equation and simultaneous approximation terms (SATs) for the weak imposition of boundary conditions and interface coupling. In particular, it focuses on operators which include boundary nodes. Of the operators considered, the hybrid-Gauss-trapezoidal-Lobatto SBP operators are the most e cient. Little di↵erence in e ciency is observed between the divergence and skew-symmetric forms, making the latter preferred given its provable stability on curved meshes. The traditional finite-di↵erence refinement strategy is the most e cient, and the discontinuous element approach the least. The continuous element refinement strategy has comparable e ciency to the traditional approach when not exhibiting lower convergence rates. This motivates a hybrid approach whereby discontinuous elements are constructed from continuous subelements. This hybrid approach is found to inherit the higher convergence rates of the traditional and discontinuous approaches, and higher e ciency relative to the discontinuous approach.

Generalized Summation by Parts Operators: Second Derivative and Time-Marching Methods

This paper describes extensions of the generalized summation-by-parts (GSBP) framework to the approximation of the second derivative with a variable coefficient and to time integration. GSBP operators for the second derivative lead to more efficient discretizations, relative to the classical finite-difference SBP approach, as they can require fewer nodes for a given order of accuracy. Similarly, for time integration, time-marching methods based on GSBP operators can be more efficient than those based on classical SBP operators, as they minimize the number of solution points which must be solved simultaneously. Furthermore, we demonstrate the link between GSBP operators and Runge-Kutta time-marching methods.

Investigation of Efficient High-Order Implicit Runge-Kutta Methods Based on Generalized Summation-by-Parts Operators

This paper summarizes several new developments in the theory of high-order implicit Runge-Kutta (RK) methods based on generalized summation-by-parts (GSBP) operators. The theory is applied to the construction of several known and novel Runge-Kutta schemes. This includes the well-known families of fully-implicit Radau IA/IIA and Lobatto IIIC Runge-Kutta methods. In addition, a novel family of GSBP-RK schemes based on Gauss quadrature rules is presented along with a few optimized diagonally-implicit GSBP-RK schemes. The novel schemes are all L-stable and algebraically stable. The stability and relative eciency of the schemes is investigated with numerical simulation of the linear convection equation with both time-independent and time-dependent convection velocities. The numerical comparison includes a few popular non-GSBP Runge-Kutta time-marching methods for reference.