Stability (learning theory)

L-stability

Differential algebraic equation

Relaxation (psychology)

Mathematics

Partial differential equation

Numerical methods for ordinary differential equations

Machine learning

Explicit and implicit methods

Mathematical analysis

Runge–Kutta methods

Psychology

Computer science

Numerical analysis

Linear multistep method

Monotonic function

Discretization

Differential equation

Applied mathematics

Ordinary differential equation

Social psychology

Mathematics of Computation

Spatial discretizations of time-dependent partial differential equations usually result in a large system of semi-linear and stiff ordinary differential equations. Taking the structures into account, we develop a family of linearly implicit and high order accurate schemes for the time discretization, using the idea of implicit-explicit Runge-Kutta methods and the relaxation techniques. The proposed schemes are monotonicity-preserving/conservative for the original problems, while the previous linearized methods are usually not. We also discuss the linear stability and strong stability preserving (SSP) property of the new relaxation methods. Numerical experiments on several typical models are presented to confirm the effectiveness of the proposed methods.

Implicit-explicit relaxation Runge-Kutta methods: construction, analysis and applications to PDEs