Attenuation effects in wave propagation are commonly modeled using visco-acoustic formulations based on the Generalized Standard Linear Solid (GSLS) framework. The accuracy of these models depends on the number of relaxation mechanisms used to approximate the desired attenuation behavior. However, each additional mechanism introduces new state variables and increases the computational cost. Existing approaches typically employ a fixed number of relaxation mechanisms throughout the computational domain, regardless of the local complexity of the wave field.
In this work, we develop a model-adaptive strategy for visco-acoustic wave propagation described by the equations $$ \varrho(x) \partial_{t}v(t,x) = \nabla p(t,{x}) + {f}(t,x) $$ $$\partial_{t} p(t,{x}) = \kappa(t,{x}) \nabla \cdot {v}(t,{x}) + \int_{0}^{t} \dot{\kappa}(t-s,{x})\, \nabla \cdot {v}(s,{x})\,\mathrm{d}s $$ with the variables velocity $v$, pressure $p$ and material parameters $\kappa$, $\varrho$ and retardation kernel $\dot{\kappa}$. In order to approximate the delayed material law of the medium we employ the GSLS model [1].
The central idea is to adjust the number of relaxation mechanisms locally in space and time according to the evolving solution. Regions not yet affected by wave propagation, or exhibiting simple wave behavior, are represented using a reduced model with fewer mechanisms. In contrast, regions containing strong heterogeneities or pronounced attenuation effects are enriched with additional relaxation mechanisms to maintain physical accuracy. This adaptive enrichment and coarsening strategy allows the computational effort to be concentrated where it is most beneficial.
A particular challenge is the presence of two independent error sources: the model error introduced by the attenuation model approximation and the discretization error arising from the numerical scheme. Reducing only one of these contributions does not necessarily improve the overall solution quality. We therefore investigate adaptive procedures that simultaneously control both errors through local refinement of the discretization and local adjustment of the attenuation model complexity. Error indicators are used to guide these decisions and to balance accuracy and efficiency.
The resulting adaptive algorithms are embedded in a high-order space-time discontinuous Galerkin (DG) discretization [2], enabling local refinement in both the numerical approximation and the attenuation model. Numerical experiments on heterogeneous benchmark problems, including the Marmousi model, demonstrate the potential of the proposed approach to substantially reduce computational costs while maintaining the accuracy required for realistic large-scale wave propagation simulations.
[1] J.O. Blanch, J.O.A. Robertsson, W.W. Symes: Modeling of a constant-Q methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique. Geophysics 1995.
[2] D.A. Ziegler: A Parallel and Adaptive Space-Time Discontinuous Galerkin Method for Visco-Elastic and Visco-Acoustic Waves. PhD thesis Karlsruhe 2020.

In this presentation, we recall the space-time POD method for linear evolution problems such as
\[
\left\{
\begin{array}{rll}
x_t + \mathcal{A} x &=\; f &\text{in } \Omega_T,\\
x &=\; 0 & \text{on } \Sigma_T,\\
x(0) &=\; x_0 & \text{in } \Omega,
\end{array}
\right.
\]
where \(\Omega\subseteq \mathbb R^d\) is open and bounded with a sufficiently smooth boundary \(\partial \Omega\), \(\Omega_T := (0,T] \times \Omega\) and \(\Sigma_T := (0,T] \times \partial \Omega\). The parabolic differential operator \(\mathcal A\) is defined by the (uniformly) continuous and coercive bilinear form \(a(t;v,w)=: \int_{\Omega_T} \mathcal{A}v \cdot w \;\mathrm{d}x \mathrm{d}t\).

The space-time POD method, as introduced in [2], is defined in the context of the tensor product spaces \(\mathcal{S}\cdot \mathcal{Y}\) for finite dimensional subspaces \(\mathcal{S}\subseteq H^1(0,T)\) and \(\mathcal{Y}\subseteq H^1_0(\Omega)\). Furthermore, the corresponding space-time POD subspace is denoted as \(\hat{\mathcal{S}}\cdot \hat{\mathcal{Y}}\) for reduced subspaces \(\hat{\mathcal{S}}\subseteq \mathcal{S}\) and \(\hat{\mathcal{Y}}\subseteq \mathcal{Y}\).
While the application of space-time POD has already been introduced in [2], we will build on those results and present an a-priori error estimate for the error between a full order solution \(x \in \mathcal{S}\cdot \mathcal{Y}\) and a reduced order solution \(\hat{x} \in \hat{\mathcal{S}}\cdot \hat{\mathcal{Y}}\) as derived in [3]. The singular value based error estimate for space-time POD follows the idea of splitting the error as
\[
\Vert x - \hat{x}\Vert_{L^2(\Omega_T)} \leq \Vert x - \mathcal{P}_{\hat{\mathcal{S}}\cdot \hat{\mathcal{Y}}} x \Vert_{L^2(\Omega_T)} + \Vert \mathcal{P}_{\hat{\mathcal{S}}\cdot \hat{\mathcal{Y}}} x - \hat{x}\Vert_{L^2(\Omega_T)} =: \Vert \varrho \Vert_{L^2(\Omega_T)} +\Vert \vartheta \Vert_{L^2(\Omega_T)},
\]
where \(\mathcal{P}_{\hat{\mathcal{S}}\cdot \hat{\mathcal{Y}}} x\) denotes the projection onto \(\hat{\mathcal{S}}\cdot \hat{\mathcal{Y}}\). While such splitting is the same as for similar a-priori error estimates for standard POD, see e.g. [1], the estimation of the error terms \(\varrho\) and \(\vartheta\) differs in the context of space-time POD.

References:

[1] - S. Banholzer, D. Beermann, L. Mechelli, and S. Volkwein, Pod suboptimal control of evolution problems: Theory and applications, 2024.
[2] - M. Baumann, P. Benner, and J. Heiland, Space-time galerkin pod with application in optimal control of semilinear partial differential equations, 2018.
[3] - C. Gräßle, J. Heiland, and J. Marquardt, A-priori error estimation for space-time Galerkin POD for linear evolution problems, 2026.

Time-dependent problems often require different resolution in different spatial regions, time intervals, and polynomial directions. Tensor-product space-time finite element methods provide a framework in which these anisotropies can be represented directly by local mesh refinement and anisotropic hp enrichment in space and time. This additional flexibility, however, leads to large coupled systems and therefore requires suitable solvers.
We present anisotropic hp-adaptive space-time finite element methods and multilevel preconditioners for the resulting linear systems. The adaptive strategy is based on goal-oriented error control in the Dual Weighted Residual framework. Directional error indicators identify whether refinement is most relevant in space, in time, or in the local polynomial degrees. This yields discretizations that adapt to goal-relevant structures without enforcing isotropic refinement.
The preconditioners exploit the Kronecker structure of tensor-product space-time discretizations. They decouple the temporal degrees of freedom for continuous and discontinuous time discretizations, leading to shifted spatial subproblems coupled through low-dimensional temporal factors. This makes the linear systems arising from anisotropic hp-adaptive space-time methods computationally feasible.
Numerical examples for convection-dominated transport problems, the heat equation, and the Stokes equations illustrate the interaction between anisotropic hp adaptivity and multilevel preconditioning.

We report on progress with numerical evolutions of the wave equation with power nonlinearity in $n+1$ dimensions. Unlike previous numerical studies, we go beyond the radial case and do not assume any symmetries for $n=3$, and we only impose an SO($n-1$) symmetry for higher dimensions. Our method is based on hyperboloidal foliations of Minkowski space and conformal compactification. Results so far include late-time power-law decay (tails) for different spherical harmonic modes, both for subcritical and supercritical, focusing and defocusing nonlinear wave equations.

Related article: Rinne O 2025 Nonlinearity 38 105026