In this paper we review and further develop a class of strong stability-preserving (SSP) high-order time discretizations for semidiscrete method of lines approximations of partial differential equations. Previously termed TVD (total variation diminishing) time discretizations, these high-order time discretization methods preserve the strong stability properties of first-order Euler time stepping and have proved very useful, especially in solving hyperbolic partial differential equations. The new developments in this paper include the construction of optimal explicit SSP linear Runge--Kutta methods, their application to the strong stability of coercive approximations, a systematic study of explicit SSP multistep methods for nonlinear problems, and the study of the SSP property of implicit Runge--Kutta and multistep methods.

/pdf/strong-stability-preserving-high-order-time-discretization-478mkrwfwr.pdf

Strong Stability-Preserving High-Order Time Discretization Methods

In this paper we further explore a class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in Shu& Osher (1988), suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD (total variation diminishing) spatial discretization, verifying the claim that TVD Runge-Kutta methods are important for such applications. We then explore the issue of optimal TVD Runge-Kutta methods for second, third and fourth order, and for low storage Runge-Kutta methods.

/pdf/total-variation-diminishing-runge-kutta-schemes-8j998a8qfs.pdf

Total variation diminishing Runge-Kutta schemes

In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate finite difference schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics.

https://www3.nd.edu/~zxu2/acms60790S13/Shu-WENO-notes.pdf

Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

In recent years, the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS) has been redesigned to better enable the use of application-specific and third-party algebraic solvers and data structures. Throughout this work, we have adhered to specific guiding principles that minimized the impact to current users while providing maximum flexibility for later evolution of solvers and data structures. The redesign was done through creation of new classes for linear and nonlinear solvers, enhancements to the vector class, and the creation of modern Fortran interfaces that leverage interoperability features of the Fortran 2003 standard. The vast majority of this work has been performed "behind-the-scenes," with minimal changes to the user interface and no reduction in solver capabilities or performance. However, these changes now allow advanced users to create highly customized solvers that exploit their problem structure, enabling SUNDIALS use on extreme-scale, heterogeneous computational architectures.

Enabling New Flexibility in the SUNDIALS Suite of Nonlinear and Differential/Algebraic Equation Solvers.

Research experience in constructing and applying higher order temporal schemes with error controllers for solving unsteady Reynolds-averaged Navier-Stokes equations is reported. The baseline flow solver under consideration uses a second-order backward difference scheme with a dual time stepping algorithm for advancing the flow solutions in time. The accuracy and efficiency of the new scheme is assessed by comparing the computational results with either analytical or highly accurate numerical solutions for aerodynamic problems of interest.

Higher order temporal schemes with error controllers for unsteady navier-stokes equations

The construction of high order entropy stable collocation schemes on quadrilateral and hexahedral elements has relied on the use of Gauss--Legendre--Lobatto collocation points and their equivalence...

Efficient Entropy Stable Gauss Collocation Methods

Implementation of the Kumaresan and Tufts algorithm to liner impedance eduction in a duct with shear flow is described. The approach is based on a noncausal model of sound propagation coupled with singular value decomposition to identify the acoustic pressure modes. The performance of the algorithm is evaluated by comparing the educed impedance spectra to that educed by a benchmark method. Results are presented using both simulated and measured data over a range of test frequencies, three mean flow Mach numbers, and six test liner structures. When simulated data are used, the impedance spectra educed is in perfect agreement with the exact impedance spectra. When measured data are used, it is found that 1) the reduced rank approximation to the prediction matrix increases the accuracy of the educed impedance, 2) the algorithm performs well except at the antiresonant and resonant frequencies of the liner, and 3) at high enough Mach number, the effects of the gradients in the mean flow boundary layer need to ...

Performance of Kumaresan and Tufts Algorithm in Liner Impedance Eduction with Flow

Staggered-grid entropy-stable multidimensional summation-by-parts discretizations on curvilinear coordinates

https://ntrs.nasa.gov/api/citations/20100002216/downloads/20100002216.pdf

Mark H. Carpenter

Papers

Higher order temporal schemes with error controllers for unsteady navier-stokes equations

Efficient Entropy Stable Gauss Collocation Methods

Performance of Kumaresan and Tufts Algorithm in Liner Impedance Eduction with Flow

Staggered-grid entropy-stable multidimensional summation-by-parts discretizations on curvilinear coordinates

Boundary closures for fourth-order energy stable weighted essentially non-oscillatory finite-difference schemes