Concepts and application of Time-limiters to high resolution schemes

• Published in: Journal of scientific computing Vol. 19; no. 1-3; pp. 139 - 162
• Main Authors: DURAISAMY, Karthikeyan, BAEDER, James D, LIU, Jian-Guo
• Format: Journal Article
• Language: English
• Published: London Plenum Publishing 01.12.2003; New York, NY Springer Nature B.V
• Subjects: Conservation equations; Discretization; Exact sciences and technology; Mathematics; Method of lines; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Ordinary differential equations; Partial differential equations, boundary value problems; Sciences and techniques of general use; Time dependence; Time integration; High resolution; Differential equation; Linear time; Numerical method; Partial differential equation; Implementation; Direct method; Time dependence; Numerical solution; Linear stability; Implicit scheme; Total variation diminishing scheme; Integration; Numerical integration; Oscillatory solution; Burgers equation; Runge Kutta method; Non linear system; Algorithm; Discretization method; Non linear stability; Equation system; Advection; Method of lines; Monotonicity
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1023/A:1025395707090

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.