# DeC and ADER: Similarities, Differences and an Unified Framework

@article{Veiga2021DeCAA, title={DeC and ADER: Similarities, Differences and an Unified Framework}, author={Maria Han Veiga and Philipp {\"O}ffner and Davide Torlo}, journal={J. Sci. Comput.}, year={2021}, volume={87}, pages={2} }

In this paper, we demonstrate that the ADER approach as it is used inter alia in [1] can be seen as a special interpretation of the deferred correction (DeC) method as introduced in [2]. By using this fact, we are able to embed ADER in a theoretical background of time integration schemes and prove the relation between the accuracy order and the number of iterations which are needed to reach the desired order. Finally, we can also investigate the stability regions for the ADER approach for…

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## References

SHOWING 1-10 OF 40 REFERENCES

### High Order Schemes for Hyperbolic Problems Using Globally Continuous Approximation and Avoiding Mass Matrices

- MathematicsJ. Sci. Comput.
- 2017

It is shown how to avoid inverting mass matrices without sacrificing the accuracy of the scheme, and the conditions under which this is possible are detailed.

### Integral deferred correction methods constructed with high order Runge-Kutta integrators

- Computer Science, MathematicsMath. Comput.
- 2010

Spectral deferred correction (SDC) methods for solving ordinary differential equations (ODEs) were introduced by Dutt, Greengard and Rokhlin (2000). It was shown in that paper that SDC methods can…

### A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes

- Computer Science, MathematicsJ. Comput. Phys.
- 2008

### ADER: Arbitrary High Order Godunov Approach

- MathematicsJ. Sci. Comput.
- 2002

The construction of non-oscillatory schemes of very high order of accuracy in space and time, to solve non-linear hyperbolic conservation laws, result from extending the ADER approach, which is related to the ENO/WENO methodology.

### Space–time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting

- Computer Science
- 2015

### Relaxation Runge-Kutta Methods: Fully-Discrete Explicit Entropy-Stable Schemes for the Euler and Navier-Stokes Equations

- Computer Science
- 2019

The framework of inner product norm preserving relaxation Runge-Kutta methods is extended to general convex quantities and conservation, dissipation, or other solution properties with respect to any convex functional are enforced by the addition of a relaxation parameter that multiplies the Runge–KutTA update at each step.

### Error Boundedness of Discontinuous Galerkin Methods with Variable Coefficients

- MathematicsJ. Sci. Comput.
- 2019

This work investigates the long time behaviour of the error of numerical solutions to time-dependent partial differential equations in the context of hyperbolic conservation laws and flux reconstruction schemes, focusing on the schemes in the discontinuous Galerkin spectral element framework.

### Towards Very High Order Godunov Schemes

- Mathematics, Computer Science
- 2001

The ADER formulation for the linear advection equation with constant coefficients, in one and multiple space dimensions, is presented and numerical results for one and two-dimensional problems using schemes of upto 10-th order accuracy are presented.

### Arbitrary high-order, conservative and positivity preserving Patankar-type deferred correction schemes

- Computer ScienceApplied Numerical Mathematics
- 2020

### ADER: A High-Order Approach for Linear Hyperbolic Systems in 2D

- PhysicsJ. Sci. Comput.
- 2002

The ADER scheme for solving systems of linear, hyperbolic partial differential equations in two-dimensions is presented and the linearised Euler equations are used for the simulation of the sound emitted by a co-rotating vortex pair.