Low-storage, Explicit Runge-Kutta Schemes for the Compressible Navier-Stokes Equations

[1]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[2]  Isaías Alonso-Mallo,et al.  Explicit single step methods with optimal order of convergence for partial differential equations , 1999 .

[3]  David W. Zingg,et al.  Runge-Kutta methods for linear ordinary differential equations , 1999 .

[4]  Charalampos Tsitouras,et al.  Cheap Error Estimation for Runge-Kutta Methods , 1999, SIAM J. Sci. Comput..

[5]  Z. Horváth,et al.  Positivity of Runge-Kutta and diagonally split Runge-Kutta methods , 1998 .

[6]  W. Habashi,et al.  2N-Storage Low Dissipation and Dispersion Runge-Kutta Schemes for Computational Acoustics , 1998 .

[7]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

[8]  D. Higham.  Regular Runge-Kutta pairs , 1997 .

[9]  D. Pathria,et al.  The Correct Formulation of Intermediate Boundary Conditions for Runge-Kutta Time Integration of Initial Boundary Value Problems , 1997, SIAM J. Sci. Comput..

[10]  O. Stein.  Bifurcations of hyperbolic fixed points for explicit Runge-Kutta methods , 1997 .

[11]  A. Bountis.  Dynamical Systems And Numerical Analysis , 1997, IEEE Computational Science and Engineering.

[12]  Desmond J. Higham,et al.  Does Error Control Suppress Spuriosity , 1997 .

[13]  F. Vadillo,et al.  On Spurious Fixed Points of Runge-Kutta Methods , 1997 .

[14]  Gordon Erlebacher,et al.  Interaction of a shock with a longitudinal vortex , 1996, Journal of Fluid Mechanics.

[15]  Manuel Calvo,et al.  Explicit Runge-Kutta methods for initial value problems with oscillating solutions , 1996 .

[16]  P. J. Prince,et al.  Continuous approximation with embedded Runge-Kutta methods , 1996 .

[17]  J. H. Verner,et al.  High-order explicit Runge-Kutta pairs with low stage order , 1996 .

[18]  Lawrence F. Shampine,et al.  An efficient Runge-Kutta (4,5) pair , 1996 .

[19]  George Papageorgiou,et al.  A family of fifth-order Runge-Kutta pairs , 1996, Math. Comput..

[20]  David W. Zingg,et al.  High-Accuracy Finite-Difference Schemes for Linear Wave Propagation , 1996, SIAM J. Sci. Comput..

[21]  David I. Gottlieb,et al.  On the Removal of Boundary Errors Caused by Runge-Kutta Integration of Nonlinear Partial Differential Equations , 1994, SIAM J. Sci. Comput..

[22]  Inge R. Gran,et al.  NEGATIVE FLAME SPEED IN AN UNSTEADY 2-D PREMIXED FLAME: A COMPUTATIONAL STUDY , 1996 .

[23]  Stephen Wolfram,et al.  The Mathematica Book , 1996 .

[24]  Emily Martin,et al.  Mathematica 3.0 standard add-on packages , 1996 .

[25]  Willem Hundsdorfer,et al.  Numerical Solution of Advection-Diffusion-Reaction Equations , 1996 .

[26]  C. David Pruett,et al.  On simulation and analysis of instability and transition in high-speed boundary-layer flows , 1995 .

[27]  David I. Gottlieb,et al.  The Theoretical Accuracy of Runge-Kutta Time Discretizations for the Initial Boundary Value Problem: A Study of the Boundary Error , 1995, SIAM J. Sci. Comput..

[28]  George Hall,et al.  A new stepsize strategy for explicit Runge-Kutta codes , 1995, Adv. Comput. Math..

[29]  C. David Pruett,et al.  Spatial direct numerical simulation of high-speed boundary-layer flows part I: Algorithmic considerations and validation , 1995 .

[30]  Desmond J. Higham,et al.  A Survey of the Explicit Runge-Kutta Method , 1995 .

[31]  M.Y. Hussaini,et al.  Low-Dissipation and Low-Dispersion Runge-Kutta Schemes for Computational Acoustics , 1994 .

[32]  Thierry Poinsot,et al.  The evolution equation for the flame surface density in turbulent premixed combustion , 1994, Journal of Fluid Mechanics.

[33]  L. Shampine,et al.  Efficiency comparisons of methods for integrating ODEs , 1994 .

[34]  Philip W. Sharp,et al.  Completely imbedded Runge-Kutta pairs , 1994 .

[35]  M. Carpenter,et al.  Several new numerical methods for compressible shear-layer simulations , 1994 .

[36]  Christopher A. Kennedy,et al.  Third-order 2N-storage Runge-Kutta schemes with error control , 1994 .

[37]  M. Carpenter,et al.  Fourth-order 2N-storage Runge-Kutta schemes , 1994 .

[38]  Kjell Gustafsson,et al.  Control-theoretic techniques for stepsize selection in implicit Runge-Kutta methods , 1991, TOMS.

[39]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[40]  Nagi N. Mansour,et al.  Compressibility effects on the growth and structure of homogeneous turbulent shear flow , 1993, Journal of Fluid Mechanics.

[41]  Philip W. Sharp,et al.  Explicit Runge-Kutta Pairs with One More Derivative Evaluation than the Minimum , 1993, SIAM J. Sci. Comput..

[42]  Marino Zennaro,et al.  Contractivity of Runge-Kutta methods with respect to forcing terms , 1993 .

[43]  J. Lambert.  Numerical Methods for Ordinary Differential Systems: The Initial Value Problem , 1991 .

[44]  Philip W. Sharp,et al.  Numerical comparisons of some explicit Runge-Kutta pairs of orders 4 through 8 , 1991, TOMS.

[45]  J. Kraaijevanger.  Contractivity of Runge-Kutta methods , 1991 .

[46]  J. H. Verner,et al.  Some Runge-Kutta formula pairs , 1991 .

[47]  J. Sowa.  Stability of a Runge-Kutta method for the Navier-Stokes equation , 1990 .

[48]  R. Marel,et al.  Stability radius of polynomials occurring in the numerical solution of initial value problems , 1990 .

[49]  B. Müller.  Linear stability condition for explicit Runge–Kutta methods to solve the compressible Navier‐Stokes equations , 1990 .

[50]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[51]  Andrew M. Stuart,et al.  Nonlinear Instability in Dissipative Finite Difference Schemes , 1989, SIAM Rev..

[52]  L. Shampine,et al.  A 3(2) pair of Runge - Kutta formulas , 1989 .

[53]  M. N. Spijker,et al.  Stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations , 1987 .

[54]  John D. Pryce,et al.  Two FORTRAN packages for assessing initial value methods , 1987, TOMS.

[55]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[56]  J. Butcher.  The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods , 1987 .

[57]  Wayne H. Enright,et al.  Interpolants for Runge-Kutta formulas , 1986, TOMS.

[58]  J. Dormand,et al.  A reconsideration of some embedded Runge-Kutta formulae , 1986 .

[59]  J. Kraaijevanger,et al.  Absolute monotonicity of polynomials occurring in the numerical solution of initial value problems , 1986 .

[60]  L. Shampine,et al.  Some practical Runge-Kutta formulas , 1986 .

[61]  Ben P. Sommeijer,et al.  Phase-Lag Analysis of Implicit Runge–Kutta Methods , 1986 .

[62]  Ben P. Sommeijer,et al.  Explicit Runge-Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions , 1987 .

[63]  J. M. Sanz-Serna,et al.  Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations , 1987 .

[64]  ScienceDirect,et al.  Applied numerical mathematics , 1985 .

[65]  J. Verwer,et al.  Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations , 1984 .

[66]  G. J. Cooper.  A Generalization of Algebraic Stability for Runge—Kutta Methods , 1984 .

[67]  M GayDavid,et al.  Algorithm 611: Subroutines for Unconstrained Minimization Using a Model/Trust-Region Approach , 1983 .

[68]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[69]  David M. author-Gay.  Computing Optimal Locally Constrained Steps , 1981 .

[70]  J. Dormand,et al.  High order embedded Runge-Kutta formulae , 1981 .

[71]  Jorge J. Moré,et al.  User Guide for Minpack-1 , 1980 .

[72]  David F. Griffiths,et al.  Analysis of error growth for explicit difference schemes in conduction–convection problems , 1980 .

[73]  J. Williamson.  Low-storage Runge-Kutta schemes , 1980 .

[74]  J. Dormand,et al.  A family of embedded Runge-Kutta formulae , 1980 .

[75]  Lawrence F. Shampine,et al.  Storage Reduction for Runge-Kutta Codes , 1979, TOMS.

[76]  R. Jeltsch,et al.  Generalized disks of contractivity for explicit and implicit Runge-Kutta methods , 1979 .

[77]  P. Houwen,et al.  On the Internal Stability of Explicit, m‐Stage Runge‐Kutta Methods for Large m‐Values , 1979 .

[78]  J. Verner.  Explicit Runge–Kutta Methods with Estimates of the Local Truncation Error , 1978 .

[79]  P. Houwen.  Explicit Runge-Kutta formulas with increased stability boundaries , 1972 .

[80]  L. Shampine.  Numerical Solution of Ordinary Differential Equations , 1995 .

[81]  Edward N. Lorenz,et al.  AN N-CYCLE TIME-DIFFERENCING SCHEME FOR STEPWISE NUMERICAL INTEGRATION , 1971 .

[82]  E. Fehlberg,et al.  Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems , 1969 .

[83]  D. Fyfe,et al.  Economical evaluation of Runge-Kutta formulae , 1966 .

[84]  J. Butcher.  On Runge-Kutta processes of high order , 1964, Journal of the Australian Mathematical Society.

[85]  J. Butcher.  Coefficients for the study of Runge-Kutta integration processes , 1963, Journal of the Australian Mathematical Society.

[86]  Samuel D. Conte,et al.  A Kutta Third-Order Procedure for Solving Differential Equations Requiring Minimum Storage , 1956, JACM.

[87]  S. Gill,et al.  A process for the step-by-step integration of differential equations in an automatic digital computing machine , 1951, Mathematical Proceedings of the Cambridge Philosophical Society.