Runge–Kutta (RK) methods may exhibit order reduction when applied to certain
stiff problems. While fully implicit RK schemes exist that avoid order reduction via
high-stage order, DIRK (diagonally implicit Runge–Kutta) schemes are practically
important due to their structural simplicity; however, these cannot possess high stage
order. The concept of weak stage order (WSO) can also overcome order
reduction, and it is compatible with the DIRK structure. DIRK schemes of
WSO up to 3 have been proposed in the past, however, they were based on
a simplified framework that cannot be extended beyond WSO 3. In this
work a general theory of WSO is employed to overcome the prior WSO
barrier and to construct practically useful high-order DIRK schemes with
WSO
and above. The resulting DIRK schemes are stiffly accurate, L-stable, have optimized
error coefficients, and are demonstrated to perform well on a portfolio of relevant
ODE and PDE test problems.
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Computer, Electrical, and
Mathematical Sciences & Engineering Division
King Abdullah University of Science and Technology
4700 KAUST
Thuwal 23955
Saudi Arabia
Computer, Electrical, and
Mathematical Sciences & Engineering Division
King Abdullah University of Science and Technology
4700 KAUST
Thuwal 23955
Saudi Arabia