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PhD thesis "Cluster-Based Parallelization of Simulations on Dynamically Adaptive Grids and Dynamic Resource Management" by Martin Schreiber.

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### 2.14 Time stepping schemes

Regarding the derivative of our conserved quantities with respect to time, we only considered the
explicit Euler method which is of first order. This section is about higher-order Runge-Kutta time
stepping schemes and sets up the basic requirements of the integration of our simulation in time. For
sake of readability, the spatial parameters ξ and η for the conserved quantities are dropped in this
section.

Typical higher-order time stepping methods such as Runge-Kutta (RK) are a commonly
chosen alternative to the explicit Euler due to their higher accuracy in time. Regarding our
requirement analysis, such higher-order methods should be obviously be considered in our
development.

With the RK method, accuracy of higher-order is achieved combining the conserved quantity
updates based on several smaller time-step computations. A generalization of the RK method
for higher-order time integration leads to s stages i = (1,2,…,s) [But64,CS01,HW08]:

yielding an explicit formulation for a_{i,j} = 0 for i < j. The solution is finally given by
Considering the framework development in Part III, we store both update values D_{i} and
conserved quantities V _{i} for each stage and finally update the conserved quantities by the
formula given with a_{i,j} and b_{i}. The coefficients a_{i,j} and b_{i} depend on the desired order of the
method and are typically given in the format of the Butcher tableau [But64] (See Appendix
A.1.6).