The development of Runge-Kutta methods for partial differential equations

DOI10.1016/0168-9274(95)00109-3zbMath0857.65094OpenAlexW2158755951MaRDI QIDQ1917459

P. J. van der Houwen

Publication date: 9 March 1997

Published in: Applied Numerical Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/0168-9274(95)00109-3

zbMATH Keywords

method of linesRunge-Kutta methods

Mathematics Subject Classification ID

Nonlinear ordinary differential equations and systems (34A34) Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations (65L06) Initial value problems for second-order parabolic equations (35K15) Method of lines for initial value and initial-boundary value problems involving PDEs (65M20)

Related Items (23)

Fractional step Runge-Kutta methods for time dependent coefficient parabolic problems ⋮ On stability and convergence of semi-Lagrangian methods for the first-order time-dependent nonlinear partial differential equations in 1D ⋮ Explicit Runge-Kutta methods for stiff problems with a gap in their eigenvalue spectrum ⋮ A class of high-order Runge-Kutta-Chebyshev stability polynomials ⋮ Accelerating moderately stiff chemical kinetics in reactive-flow simulations using GPUs ⋮ A stabilized Runge-Kutta-Legendre method for explicit super-time-stepping of parabolic and mixed equations ⋮ Second-order accurate projective integrators for multiscale problems ⋮ Time stepping via one-dimensional Padé approximation ⋮ Implementation of low-storage Runge-Kutta time integration schemes in scalable asynchronous partial differential equation solvers ⋮ Many-stage optimal stabilized Runge-Kutta methods for hyperbolic partial differential equations ⋮ Explicit finite difference schemes with extended stability for advection equations ⋮ Towards explicit methods for differential algebraic equations ⋮ High-performance implementation of a Runge-Kutta finite-difference scheme for the Higgs boson equation in the de Sitter spacetime ⋮ A time splitting scheme for the nonhydrostatic atmospheric model ⋮ Mathematical home burglary model with stochastic long crime trips and patrolling: applied to Mexico City ⋮ Kinetic structures in multi-phase liquid-crystalline composite materials ⋮ On iterated Crank-Nicolson methods for hyperbolic and parabolic equations ⋮ Stability results for fractional step discretizations of time dependent coefficient evolutionary problems ⋮ Time integration of the shallow water equations in spherical geometry ⋮ Stability results for linearly implicit fractional step discretizations of nonlinear time dependent parabolic problems ⋮ Explicit Runge-Kutta methods for parabolic partial differential equations ⋮ Optimized low-dispersion and low-dissipation two-derivative Runge-Kutta method for wave equations ⋮ Additive Runge-Kutta methods for the resolution of linear parabolic problems

Uses Software

• RKC

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• One step integration methods with maximum stability regions
• One step integration methods of third-fourth order accuracy with large hyperbolic stability limits
• Convergence properties of the Runge-Kutta-Chebyshev method
• New stability theorems concerning one-step numerical methods for ordinary differential equations
• Order and stability properties of explicit multivalue methods
• Stability of explicit time discretizations for solving initial value problems
• Stability and accuracy of time discretizations for initial value problems
• One-step splitting methods for semi-discrete parabolic equations
• Splitting methods for three-dimensional transport models with interaction terms
• Explicit Runge-Kutta formulas with increased stability boundaries
• Optimal stability polynomials
• The Numerical Solution of Parabolic and Elliptic Differential Equations
• Numerical Stability in Digital and Analog Computation for Diffusion Problems
• On the Internal Stability of Explicit,m-Stage Runge-Kutta Methods for Largem-Values
• A Special Class of Multistep Runge—Kutta Methods with Extended Real Stability Interval
• The automatic integration of ordinary differential equations
• A New Computational Procedure for A.D.I. Methods
• Integration of Stiff Equations
• On Richardson's Method for Solving Linear Systems with Positive Definite Matrices

This page was built for publication: The development of Runge-Kutta methods for partial differential equations