A Priori Time Limiting for the Quinpi Scheme

Stiff hyperbolic problems can be challenging for numerical methods to solve. The Quinpi numerical method uses an implicit third-order Central Weighted Essentially Non-Oscillatory space reconstruction and a third-order Diagonally Implicit Runge-Kutta scheme for the time integration. Backward Euler predictors are employed to ease the solution of the nonlinear equations of the DIRK scheme. Despite limiting the solution in space, spurious oscillations can occur at large time steps. Therefore, several a posteriori techniques were developed to reduce the oscillations. In this work, an a priori technique is presented, where the oscillations are reduced using a weighting procedure inspired by spatial WENO methods. The weights combine the high-order Runge-Kutta scheme with the composite Backward Euler scheme. A smoothness indicator is required to compute the weights. The proposed indicator is based on the numerical flux differences multiplied by the Runge-Kutta coefficients. The a priori technique is applied and compared on the scalar linear and nonlinear conservation laws in one-dimensional and two-dimensional space, and also on the linear and nonlinear systems of equations in one-dimensional space.