The significance of temporal discretization in the high order numerical scheme for hyperbolic conservation law

Spatial and temporal discretization has a big impact to the overall accuracy of the numerical scheme. The spatial discretization affects the calculation of physical flux or other space-derived variables, which often require high order scheme in the non-linear problem to achieve the desired accuracy. While temporal discretization also has a significant impact on the stability of the numerical scheme and total computational cost. Strong stability preserving Runge Kutta scheme often used as temporal discretization scheme for high order numerical scheme. In this work, we are motivated to study the significance of temporal discretization order (up to fourth order) to the overall numerical scheme accuracy and total computational cost while high order weighted essentially non-oscillatory scheme used as spatial discretization technique in hyperbolic conservation law. We found that the third order SSP Runge Kutta scheme is an effective scheme with respect to accuracy and computational cost.