Uniformly high-order accurate nonoscillatory schemes
SIAM Journal on Numerical Analysis
Accurate monotone cubic interpolation
SIAM Journal on Numerical Analysis
An Accurate and Robust Flux Splitting Scheme for Shock and Contact Discontinuities
SIAM Journal on Scientific Computing
Accurate monotonicity-preserving schemes with Runge-Kutta time stepping
Journal of Computational Physics
Total variation diminishing Runge-Kutta schemes
Mathematics of Computation
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A new class of implicit high-order non-oscillatory time integration schemes is introduced in a method-of-lines framework. These schemes can be used in conjunction with an appropriate spatial discretization scheme for the numerical solution of time dependent conservation equations. The main concept behind these schemes is that the order of accuracy in time is dropped locally in regions where the time evolution of the solution is not smooth. By doing this, an attempt is made at locally satisfying monotonicity conditions, while maintaining a high order of accuracy in most of the solution domain. When a linear high order time integration scheme is used along with a high order spatial discretization, enforcement of monotonicity imposes severe time-step restrictions. We propose to apply limiters to these time-integration schemes, thus making them non-linear. When these new schemes are used with high order spatial discretizations, solutions remain non-oscillatory for much larger time-steps as compared to linear time integration schemes. Numerical results obtained on scalar conservation equations and systems of conservation equations are highly promising.