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
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
Diagonally-implicit multi-stage integration methods
Applied Numerical Mathematics
An implicit upwind algorithm for computing turbulent flows on unstructured grids
Computers and Fluids
Control-theoretic techniques for stepsize selection in implicit Runge-Kutta methods
ACM Transactions on Mathematical Software (TOMS)
Control theoretic techniques for stepsize selection in explicit Runge-Kutta methods
ACM Transactions on Mathematical Software (TOMS)
Implicit-explicit methods for time-dependent partial differential equations
SIAM Journal on Numerical Analysis
Runge-Kutta research in Trondheim
Applied Numerical Mathematics - Special issue celebrating the centenary of Runge-Kutta methods
Control Strategies for the Iterative Solution of Nonlinear Equations in ODE Solvers
SIAM Journal on Scientific Computing
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations
Applied Numerical Mathematics - Special issue on time integration
Linearly implicit Runge—Kutta methods for advection—reaction—diffusion equations
Applied Numerical Mathematics
Additive Runge-Kutta schemes for convection-diffusion-reaction equations
Applied Numerical Mathematics
Journal of Computational Physics
An Efficient Fourth Order Implicit Runge-Kutta Algorithm for Second Order Systems
Computer Mathematics
Fast unsteady flow computations with a Jacobian-free Newton-Krylov algorithm
Journal of Computational Physics
Implicit-explicit schemes for flow equations with stiff source terms
Journal of Computational and Applied Mathematics
Marker Redistancing/Level Set Method for High-Fidelity Implicit Interface Tracking
SIAM Journal on Scientific Computing
A framework for parallel computational physics algorithms on multi-core: SPH in parallel
Advances in Engineering Software
Journal of Computational Physics
Hi-index | 0.02 |
Multiple high-order time-integration schemes are used to solve stiff test problems related to the Navier---Stokes (NS) equations. The primary objective is to determine whether high-order schemes can displace currently used second-order schemes on stiff NS and Reynolds averaged NS (RANS) problems, for a meaningful portion of the work-precision spectrum. Implicit---Explicit (IMEX) schemes are used on separable problems that naturally partition into stiff and nonstiff components. Non-separable problems are solved with fully implicit schemes, oftentimes the implicit portion of an IMEX scheme. The convection---diffusion-reaction (CDR) equations allow a term by term stiff/nonstiff partition that is often well suited for IMEX methods. Major variables in CDR converge at near design-order rates with all formulations, including the fourth-order IMEX additive Runge---Kutta (ARK2) schemes that are susceptible to order reduction. The semi-implicit backward differentiation formulae and IMEX ARK2 schemes are of comparable efficiency. Laminar and turbulent aerodynamic applications require fully implicit schemes, as they are not profitably partitioned. All schemes achieve design-order convergence rates on the laminar problem. The fourth-order explicit singly diagonally implicit Runge---Kutta (ESDIRK4) scheme is more efficient than the popular second-order backward differentiation formulae (BDF2) method. The BDF2 and fourth-order modified extended backward differentiation formulae (MEBDF4) schemes are of comparable efficiency on the turbulent problem. High precision requirements slightly favor the MEBDF4 scheme (greater than three significant digits). Significant order reduction plagues the ESDIRK4 scheme in the turbulent case. The magnitude of the order reduction varies with Reynolds number. Poor performance of the high-order methods can partially be attributed to poor solver performance. Huge time steps allowed by high-order formulations challenge the capabilities of algebraic solver technology.