The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
Total-variation-diminishing time discretizations
SIAM Journal on Scientific and Statistical Computing
Numerical methods for ordinary differential systems: the initial value problem
Numerical methods for ordinary differential systems: the initial value problem
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Weighted essentially non-oscillatory schemes
Journal of Computational Physics
Finite difference schemes for long-time integration
Journal of Computational Physics
A positive finite-difference advection scheme
Journal of Computational Physics
The Runge-Kutta Theory in a Nutshell
SIAM Journal on Numerical Analysis
LOW-DISSIPATION AND -DISPERSION RUNGE-KUTTA SCHEMES FOR COMPUTATIONAL ACOUSTICS
LOW-DISSIPATION AND -DISPERSION RUNGE-KUTTA SCHEMES FOR COMPUTATIONAL ACOUSTICS
A family of low dispersive and low dissipative explicit schemes for flow and noise computations
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (Classics in Applied Mathematics Classics in Applied Mathemat)
Conditionally Stable Multidimensional Schemes for Advective Equations
Journal of Scientific Computing
A PML-based nonreflective boundary for free surface fluid animation
ACM Transactions on Graphics (TOG)
Journal of Computational Physics
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For the 1-dim. linear advection problem stability limits of Runge-Kutta (RK) methods from 1st to 7th order in combination with upwind or centered difference schemes from 1st to 6th order are presented. The analysis can be carried out in a rather general way by introduction of a broad class of Runge-Kutta methods, here called 'Linear Case Runge-Kutta (LC-RK)' methods, which behave completely similar for linear, time-independent and homogeneous ODE-systems and contain the 'classical' order=stage RK methods. The set of conditions for the coefficients of these LC-RK-schemes could be derived explicitly for arbitrary order N. From an efficiency viewpoint the LC-RK 3rd order methods in combination with upwind 3rd or 5th order or the LC-RK 4th order scheme with 4th order centered difference advection are a good choice. The analysis can be extended easily to multidimensional splited advection for which a necessary stability condition is presented.