Overcoming order reduction in Runge-Kutta methods via weak stage order: Theory and order barriers
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When Runge-Kutta (RK) methods are applied as time-discretizations of partial differential equations, they may result in a loss of accuracy otherwise known as order reduction. In this talk we present recent work on avoiding order reduction through the addition of extra stiff order conditions designed specifically for linear, and semilinear problems. These order conditions are less restrictive than traditional stage order conditions and are compatible with a practically important DIRK structure. We will then present a recently developed theory establishing order barriers for the stiff order conditions, construct schemes that satisfy the barriers sharply, and test the schemes on PDE problems demonstrating their efficacy. The key mathematical ideas make use of pairs of orthogonal invariant subspaces, an embedded Sylvester equation, and reformulations of reducible RK schemes in terms of invariant subspaces.