For time-dependent PDE, the most common temporal discretization is a time-marching approach in which a uniform timestep is taken over the entire spatial domain. Consider the propagation of a compact wave with size *l* over a distance *L.* Suppose that for a given discretization that the number of spatial elements to adequately represent the wave is *N *and that the number of timesteps is a constant multiple of *N*. Then, to accurately simulate the propagation of the wave a distance *L * using a uniform mesh in space and time would require *O((L/l)^2) *degrees of freedom.

An alternative strategy is to locally refine only in the vicinity of the wave but require the discretization to still use a time-marching approach. We refer to this as a space-time tensor-product adaptive refinement as the spatial and temporal adaptations are performed independently. This tensor-product adaptive refinement method is essentially the approach known as Adaptive Mesh Refinement (AMR) and generally utilizes Cartesian hanging-node spatial meshes. AMR is among the most widely used methods for wave propagation problems. Still, the cost to adequately resolve the solution depends on scale ratio, specifically, requiring *O(L/l) *degrees of freedom.

Another alternative is to allow general adaptation in space and time and we refer to this approach as a space-time unstructured approach. With this approach, the degrees of freedom required will now be independent of the scale ratio; that is the degrees of freedom are *O(1)* as *L/l *increases. Thus, the degree of freedoms required is robust to the ratio of the wave and domain length scales. We make the following observations about a general unstructured space-time adaptive approach:

- A potential downside to this approach is that all degrees of freedom are coupled as opposed to a time-marching method which is only solve problems for the next time level.
- An interesting possibility is that the need to solve a global space-time problem may allow better parallelization since a time-marching approach has a natural serial barrier due to the temporal casuality. This possibility is at the heart of so-called parallel-in-time methods.
- As the problems of interests involve 3D spatial domains, the realization of a space-time unstructured adaptive method requires the ability to generate unstructured meshes in 4D (i.e. 3D + time).

**Example: 2D vortex convection**

Yano[1] demonstrated the application of space-time adaptation using the MOESS algorithm. In this example, an inviscid compressible (though low Mach number) vortical flow is considered. The density distribution at the initial (*t=0*) and final (*t=20*) times are shown, as well as in the *x _{2}=0 *space-time plane. The adaptive mesh is also shown, with the MOESS adaptation based upon controlling the (output) error in the momentum perturbation at the final time.

The convergence plot below compares the error for uniformly refined and space-time adapted meshes using *p=1-3 * polynomial elements. We observe the significant gains possible using the space-time adaptive method. For example, to achieve 0.1% accuracy requires about 20 times more DOF for uniform *p=3 * meshes compared to the adapted meshes. We also note that the space-time adaptive method converges at a faster rate, indicating that the MOESS adaptive method has reduce the dimensionality of the problem from a fully 3D problem to one which is effectively 2D.

**Example: 2D reservoir simulation**

Jayasinghe studied the use of space-time adaptation for oil reservoir simulations[2][3][4] An example 2D problem is the simulation of a slightly compressible water-oil system driven by a two-spot well configuration in a square, heterogeneous reservoir. In particular, the heterogeneity involves a region with a factor of 100 less permeability than the majority of the reservoir. The injector and producer are modeled[5] with fixed bottom hole pressures as shown in the figure, and the resulting pressure differential drives the flow from the injector towards the producer.

The space-time adaptive simulation is performed using the MOESS algorithm with the output of interest being the oil recovery factor. The resulting space-time adapted mesh for a discontinuous Galerkin (DG) FEM with linear elements is shown in the figure with the water saturation levels. The adaptive algorithm focuses on the near-well regions with more anisotropic resolution around the production well than around the injection well. The algorithm also automatically targets the boundaries of the heterogeneous block, without any prior knowledge of its existence, in order to accurate capture the permeability discontinuities.

The convergence of the oil recovery factor predictions with mesh refinement for a variety of different approaches is shown below including Finite Volume (FV) and DG methods, with both time-marching on uniform quadrilateral spatial elements as well as the space-time adapted DG approach. The abscissa represents the total number of space-time unknowns in each solution, given by (N_{DOF }N_{state}), where N_{DOF} is the number of space-time degrees-of-freedom per state variable, and N_{state} is the number of state variables used by the discretization. For the finite volume method which only solves for pressure and saturation, N_{state} = 2, whereas for the DG methods N_{state} = 3, since they also solve for artificial viscosity. However, even with the additional degrees-of-freedom associated with artificial viscosity, the linear (P1) and quadratic (P2) adaptive space-time DG methods (solid red and green lines) only require about one million unknowns to achieve an error tolerance of ±0.1% (marked by the dotted horizontal lines). In contrast, the time-marching finite volume method requires about one billion unknowns to achieve the same error level. The time-marching DG methods and the space-time DG methods on structured tetrahedral meshes fare inbetween the two extremes discussed above.

**Example: 4D metric-based adaptive meshing**

Caplan[6] in his doctoral work developed an anisotropic metric-based adaptive mesher for 4-dimensional problems. This is a key requirement to utilize a space-time adaptive approach on time-dependent problems with three-spatial dimensions. An example of a 4D mesh for a solution exhibiting an expanding spherical wave front is shown in the figure below. Note that the figure shows only the surface mesh on the planar boundaries of the 4D hypercube domain. General visualization techniques for 4D meshes and solutions still remain to be developed.

### References

- Yano M. An Optimization Framework for Adaptive Higher-Order Discretizations of Partial Differential Equations on Anisotropic Simplex Meshes [PhD]. Department of Aeronautics and Astronautics: Massachusetts Institute of Technology; 2012. Available from: http://hdl.handle.net/1721.1/96928
- Jayasinghe S. An Adaptive Space-Time Discontinuous Galerkin Method for Reservoir Flows [PhD]. Department of Aeronautics and Astronautics: Massachusetts Institute of Technology; 2018.
- Jayasinghe YS, Darmofal DL, Burgess NK, Galbraith MC, Allmaras SR. A space-time adaptive method for reservoir flows: Formulation and one-dimensional application. Computational Geosciences [Internet]. 2018 ;22(1):107-123. Available from: http://dx.doi.org/10.1007/s10596-017-9673-9
- Jayasinghe YS. A Space-time Adaptive Method for Flows in Oil Reservoirs [masters]. Department of Aeronautics and Astronautics: Massachusetts Institute of Technology; 2015. Available from: http://hdl.handle.net/1721.1/101495
- Jayasinghe S, Darmofal DL, Dow E, Galbraith MC, Allmaras SR. A discretization-independent distributed well model. Society of Petroleum Engineers Journal. 2019 .
- Caplan PC. Four-dimensional Anisotropic Mesh Adaptation for Spacetime Numerical Simulations [PhD]. Department of Aeronautics and Astronautics: Massachusetts Institute of Technology; 2019.