Parallel-in-Time
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+StroemungsRaum
+==============
+
+**StroemungsRaum** is a research software project developed within the
+BMBF-funded project
+
+*“StrömungsRaum – Novel Exascale Architectures with Heterogeneous Hardware
+Components for Computational Fluid Dynamics Simulations”*
+(October 2022 – September 2025).
+
+The project addresses the development of scalable numerical methods and
+high-performance algorithms for Computational Fluid Dynamics (CFD) targeting
+future **exascale computing architectures** with heterogeneous hardware.
+
+Scope of This Repository
+------------------------
+This repository contains the **Forschungszentrum Jülich (FZJ)** contribution to
+the StrömungsRaum project, focusing on:
+
+- Parallel-in-time methods
+- Combined space–time parallelization for fluid simulations
+- Algorithmic scalability for time-dependent PDEs
+
+The goal is to expose concurrency beyond spatial parallelism and enable
+efficient execution on large-scale HPC systems.
+
+Model Problems and Methods
+--------------------------
+Implemented examples and test cases include:
+
+- Heat equation
+- Convection–diffusion and nonlinear convection–diffusion problems
+- Incompressible Navier–Stokes equations, using:  
+   - Projection methods
+   - Monolithic formulations
+   - DAE- and PDE sweepers
+
+These serve as benchmarks and demonstrators for scalable space–time CFD
+simulations.
+
+Funding
+-------
+Funded by the **German Federal Ministry of Education and Research (BMBF)** under
+grant number **16ME0708**.
+
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+---
+
+name: pySDC
+channels:
+  - conda-forge
+dependencies:
+  - fenics>=2019.1.0
+  - numpy
+  - scipy>=0.17.1
+  - dill>=0.2.6
+  - matplotlib>=3.0
+  - pytest
+  - pytest-cov
+  - pip
+  - pip:
+    - qmat>=0.1.8
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+import logging
+
+import dolfin as df
+import numpy as np
+
+from pySDC.core.problem import Problem
+from pySDC.implementations.datatype_classes.fenics_mesh import fenics_mesh, rhs_fenics_mesh
+
+
+class fenics_heat2D_mass(Problem):
+    r"""
+    Example implementing the forced two-dimensional heat equation with Dirichlet boundary conditions
+
+    .. math::
+        \frac{\partial u}{\partial t} = \nu \Delta u + f
+
+    for :math:`x \in \Omega:=[0,1]x[0,1]`, where the forcing term :math:`f` is defined by
+
+    .. math::
+        f(x, y, t) = -\sin(\pi x)\sin(\pi y) (\sin(t) - 2\nu \pi^2 \cos(t)).
+
+    For initial conditions with constant c and
+
+    .. math::
+        u(x, y, 0) = \sin(\pi x)\sin(\pi y) + c
+
+    the exact solution of the problem is given by
+
+    .. math::
+        u(x, y, t) = \sin(\pi x)\sin(\pi y)\cos(t) + c.
+
+    In this class the spatial part is solved using ``FEniCS`` [1]_. Hence, the problem is reformulated to
+    the *weak formulation*
+
+    .. math:
+        \int_\Omega u_t v\,dx = - \nu \int_\Omega \nabla u \nabla v\,dx + \int_\Omega f v\,dx.
+
+    The part containing the forcing term is treated explicitly, where it is interpolated in the function
+    space. The other part will be treated in an implicit way.
+
+    Parameters
+    ----------
+    c_nvars : int, optional
+        Spatial resolution.
+    t0 : float, optional
+        Starting time.
+    family : str, optional
+        Indicates the family of elements used to create the function space
+        for the trail and test functions. The default is ``'CG'``, which are the class
+        of Continuous Galerkin, a *synonym* for the Lagrange family of elements, see [2]_.
+    order : int, optional
+        Defines the order of the elements in the function space.
+    nu : float, optional
+        Diffusion coefficient :math:`\nu`.
+    c: float, optional
+        Constant for the Dirichlet boundary condition :math: `c`
+
+    Attributes
+    ----------
+    V : FunctionSpace
+        Defines the function space of the trial and test functions.
+    M : scalar, vector, matrix or higher rank tensor
+        Denotes the expression :math:`\int_\Omega u_t v\,dx`.
+    K : scalar, vector, matrix or higher rank tensor
+        Denotes the expression :math:`- \nu \int_\Omega \nabla u \nabla v\,dx`.
+    g : Expression
+        The forcing term :math:`f` in the heat equation.
+    bc : DirichletBC
+        Denotes the Dirichlet boundary conditions.
+
+    References
+    ----------
+    .. [1] The FEniCS Project Version 1.5. M. S. Alnaes, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg,
+        C. Richardson, J. Ring, M. E. Rognes, G. N. Wells. Archive of Numerical Software (2015).
+    .. [2] Automated Solution of Differential Equations by the Finite Element Method. A. Logg, K.-A. Mardal, G. N.
+        Wells and others. Springer (2012).
+    """
+
+    dtype_u = fenics_mesh
+    dtype_f = rhs_fenics_mesh
+
+    def __init__(self, c_nvars=64, t0=0.0, family='CG', order=2, nu=0.1, c=0.0):
+
+        # define the Dirichlet boundary
+        def Boundary(x, on_boundary):
+            return on_boundary
+
+        # set solver and form parameters
+        df.parameters["form_compiler"]["optimize"] = True
+        df.parameters["form_compiler"]["cpp_optimize"] = True
+        df.parameters['allow_extrapolation'] = True
+
+        # set mesh and refinement (for multilevel)
+        mesh = df.UnitSquareMesh(c_nvars, c_nvars)
+
+        # define function space for future reference
+        self.V = df.FunctionSpace(mesh, family, order)
+        tmp = df.Function(self.V)
+        self.logger.debug('DoFs on this level:', len(tmp.vector()[:]))
+
+        # invoke super init, passing number of dofs, dtype_u and dtype_f
+        super().__init__(self.V)
+        self._makeAttributeAndRegister('c_nvars', 't0', 'family', 'order', 'nu', 'c', localVars=locals(), readOnly=True)
+
+        # Stiffness term (Laplace)
+        u = df.TrialFunction(self.V)
+        v = df.TestFunction(self.V)
+        a_K = -1.0 * df.inner(df.nabla_grad(u), self.nu * df.nabla_grad(v)) * df.dx
+
+        # Mass term
+        a_M = u * v * df.dx
+
+        self.M = df.assemble(a_M)
+        self.K = df.assemble(a_K)
+
+        # set boundary values
+        self.u_D = df.Expression('sin(a*x[0]) * sin(a*x[1]) * cos(t) + c', c=self.c, a=np.pi, t=t0, degree=self.order)
+        self.bc = df.DirichletBC(self.V, self.u_D, Boundary)
+        self.bc_hom = df.DirichletBC(self.V, df.Constant(0), Boundary)
+
+        self.fix_bc_for_residual = True
+
+        # set forcing term as expression
+        self.g = df.Expression(
+            '-sin(a*x[0]) * sin(a*x[1]) * (sin(t) - 2*b*a*a*cos(t))',
+            a=np.pi,
+            b=self.nu,
+            t=self.t0,
+            degree=self.order,
+        )
+
+    def solve_system(self, rhs, factor, u0, t):
+        r"""
+        Dolfin's linear solver for :math:`(M - factor A) \vec{u} = \vec{rhs}`.
+
+        Parameters
+        ----------
+        rhs : dtype_f
+            Right-hand side for the nonlinear system.
+        factor : float
+            Abbrev. for the node-to-node stepsize (or any other factor required).
+        u0 : dtype_u
+            Initial guess for the iterative solver (not used here so far).
+        t : float
+            Current time.
+
+        Returns
+        -------
+        u : dtype_u
+            Solution.
+        """
+
+        u = self.dtype_u(u0)
+        T = self.M - factor * self.K
+        b = self.dtype_u(rhs)
+
+        self.u_D.t = t
+
+        self.bc.apply(T, b.values.vector())
+        self.bc.apply(b.values.vector())
+
+        df.solve(T, u.values.vector(), b.values.vector())
+
+        return u
+
+    def eval_f(self, u, t):
+        """
+        Routine to evaluate both parts of the right-hand side.
+
+        Parameters
+        ----------
+        u : dtype_u
+            Current values of the numerical solution.
+        t : float
+            Current time at which the numerical solution is computed.
+
+        Returns
+        -------
+        f : dtype_f
+            The right-hand side divided into two parts.
+        """
+
+        f = self.dtype_f(self.V)
+
+        self.K.mult(u.values.vector(), f.impl.values.vector())
+
+        self.g.t = t
+        f.expl = self.apply_mass_matrix(self.dtype_u(df.interpolate(self.g, self.V)))
+        return f
+
+    def apply_mass_matrix(self, u):
+        r"""
+        Routine to apply mass matrix.
+
+        Parameters
+        ----------
+        u : dtype_u
+            Current values of the numerical solution.
+
+        Returns
+        -------
+        me : dtype_u
+            The product :math:`M \vec{u}`.
+        """
+
+        me = self.dtype_u(self.V)
+        self.M.mult(u.values.vector(), me.values.vector())
+
+        return me
+
+    def u_exact(self, t):
+        r"""
+        Routine to compute the exact solution at time :math:`t`.
+
+        Parameters
+        ----------
+        t : float
+            Time of the exact solution.
+
+        Returns
+        -------
+        me : dtype_u
+            Exact solution.
+        """
+        u0 = df.Expression('sin(a*x[0]) * sin(a*x[1]) * cos(t) + c', c=self.c, a=np.pi, t=t, degree=self.order)
+        me = self.dtype_u(df.interpolate(u0, self.V), val=self.V)
+
+        return me
+
+    def fix_residual(self, res):
+        """
+        Applies homogeneous Dirichlet boundary conditions to the residual
+
+        Parameters
+        ----------
+        res : dtype_u
+              Residual
+        """
+        self.bc_hom.apply(res.values.vector())
+        return None