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A Three-Level Linearized Time Integration Scheme for Tumor Simulations with Cahn-Hilliard Equations

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Computational Science – ICCS 2021 (ICCS 2021)

Abstract

The paper contains an analysis of a three-level linearized time integration scheme for Cahn-Hilliard equations. We start with a rigorous mixed strong/variational formulation of the appropriate initial boundary value problem taking into account the existence and uniqueness of its solution. Next we pass to the definition of two time integration schemes: the Crank-Nicolson and a three-level linearized ones. Both schemes are applied to the discrete version of Cahn-Hilliard equation obtained through the Galerkin approximation in space. We prove that the sequence of solutions of the mixed three level finite difference scheme combined with the Galerkin approximation converges when the time step length and the space approximation error decrease. We also recall the verification of the second order of this scheme and its unconditional stability with respect to the time variable. A comparative scalability analysis of parallel implementations of the schemes is also presented.

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Acknowledgement

The Authors are thankful for support from the funds assigned to AGH University of Science and Technology by the Polish Ministry of Science and Higher Education.

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Correspondence to Maciej Woźniak .

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Appendix: Convergence of the Mixed 3-level Linearized Scheme

Appendix: Convergence of the Mixed 3-level Linearized Scheme

In this section we prove the convergence of the 3-level scheme (13) . Crucial properties of operator A are its continuity and coercivity, which are used to prove the convergence of the numerical schema. They are formulated in the following way: there exist positive constants m and M and function \(\zeta \) satisfying \(\zeta (s) \rightarrow +\infty (s \rightarrow +\infty )\) such that for every \(u, v \in V\) we have

$$\begin{aligned} \Vert A(u) - A(v) \Vert _{V'} \le M \Vert u - v \Vert _V \end{aligned}$$
(14a)
$$\begin{aligned} \langle A(u) - A(v), u - v \rangle _{V' \times V} \ge m \Vert u -v \Vert ^2_V \end{aligned}$$
(14b)
$$\begin{aligned} \langle A(u), u \rangle _{V' \times V} \ge \zeta (\Vert u\Vert _V) \ \Vert u\Vert _V \end{aligned}$$
(14c)

A sample case when the above conditions hold is shown in the following observation.

Observation 3

Assume that both B and \(\varPsi ''\) are positive constants (\(B \equiv b > 0\) and \(\varPsi '' \equiv c > 0\)). Then, conditions () hold.

Proof

This is in fact a linear case, i.e.

$$ \langle A(u), w \rangle = \int _\varOmega \left( \gamma b \varDelta u \varDelta w + b c \nabla u \nabla w \right) dx. $$

Therefore,

$$ | \langle A(u), w \rangle | \le \gamma b \Vert \varDelta u \Vert _{L^2(\varOmega )} \Vert \varDelta w \Vert _{L^2(\varOmega )} + b c \Vert \nabla u \Vert _{L^2(\varOmega ;\mathbb {R}^n)} \Vert \nabla w \Vert _{L^2(\varOmega ;\mathbb {R}^n)}, $$

which yields (14a) with, e.g., \(M = b (\gamma + c)\). Moreover,

$$\begin{aligned} \langle A(u), u \rangle&= \int _\varOmega \left( \gamma b (\varDelta u)^2 + b c |\nabla u|^2 \right) dx\\&= \gamma b \Vert \varDelta u \Vert ^2_{L^2(\varOmega )} + b c \Vert \nabla u \Vert ^2_{L^2(\varOmega ;\mathbb {R}^n)}, \end{aligned}$$

which, together with an appropriate version of Poincaré inequality, gives us (14c). Finally, in this case, it is easy to see that (14b) is a consequence of (14c).

Let assume now, that we know the solution to (10) in some interval \([-T_l, 0]\) for \(T_l > 0\). We can introduce the time grids

$$\begin{aligned} S_\tau = \{ i \tau ; \, i = 1, 2, \ldots , K; \, \tau < t_0, \, K \tau = T\}. \end{aligned}$$
(15)

The arbitrary function \(g: [-T_l, T] \rightarrow X_n ((X_n)')\) can be restricted to \(S_\tau \), then we obtain the grid function \(g_\tau = \{g_\tau ^{-1} = g(-\tau ), g_\tau ^0 = g(0), g_\tau ^1 = g(\tau ), g_\tau ^2 = g(2 \tau ), \ldots , g_\tau ^K = g(K \tau )\}\).

Let us denote by \(V_n\) and \(V'_n\) the vector spaces being the the restrictions of \(C(-T_l,T; X_n)\) and \(C(-T_l,T; X'_n)\) to the network \(S_\tau \) equipped with the norms:

(16)

respectively.

We are ready now to define the time grid operator \(R_\tau : V_n \rightarrow V'_n\) as the collection of coordinate operators

$$\begin{aligned} (R_\tau (g_\tau ))^i = \frac{g_\tau ^{i+1} - g_\tau ^i}{2 \tau } + A(g_\tau ^i) + \frac{1}{2} DA|_{g_\tau ^{i+1}} (g_\tau ^{i+2} - 2 g_\tau ^i + g_\tau ^{i-1}) \end{aligned}$$
(17)

associated with the three-level linearized scheme (13). The mixed Galerkin three-level linearized scheme discrete problem can be formulated as follows:

Let us assume, that the exact solution u of (10) is well-known and continuous with respect to the time variable on the interval \([-T_l, 0]\) and moreover \(\forall t \in [-T_l, 0] \; u(t) \in \bigcap _n X_n\). We are looking for \(u_{n\tau } \in V_n\) that satisfies

$$\begin{aligned} R_\tau (u_{n\tau }) = 0 \; \; \; \text {and} \; \; \; u_{n\tau }^{-1} = u(-\tau ), \; u_{n\tau }^0 = u_0. \end{aligned}$$
(18)

Notice, that for the sake of simplicity the notation of the operator \(R_\tau \) is polymorphic in the same way as the notation of A here, i.e. denote the families of operators for all n.

Solving nonlinear parabolic variational equations of type (10) by using mixed Galerkin 3-leveled linearized schema was intensively studied in papers [26, 27]. Observation 3 states the fact that operator A satisfies the assumptions of Theorems 1 and 2 from paper [27]. In particular, Theorem 1 in [27] implies that:

Observation 4

Under the assumptions of Observation 3 the following statements hold:

  1. 1.

    Problem (10) has the unique solution in \(L^2(0,T; V) \cap C(0,T;L^2(\varOmega ))\) for any \(u_0\).

  2. 2.

    Problem (11) has the unique solution in \(C^1(0,T;X_n)\) for any \(u_0\) and n.

  3. 3.

    \(\left\| u_n - u \right\| _{C(0,T;L^2(\varOmega ))}\) for \(n \rightarrow +\infty \).

Moreover, taking into account Theorem 2 in [27] we have:

Observation 5

The problem (18) has the unique solution for any \(u(-\tau )\) and \(u_0\), moreover

$$\lim _{n \rightarrow +\infty , \tau \rightarrow 0} \left\| u_{n\tau } - u \right\| _\tau = 0$$

where \(\left\| u\right\| _\tau = \max \{\left\| u(i\tau )\right\| _{L^2(\varOmega )}, i = 1,2, \ldots , K\}\).

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Smołka, M., Woźniak, M., Schaefer, R. (2021). A Three-Level Linearized Time Integration Scheme for Tumor Simulations with Cahn-Hilliard Equations. In: Paszynski, M., Kranzlmüller, D., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M.A. (eds) Computational Science – ICCS 2021. ICCS 2021. Lecture Notes in Computer Science(), vol 12747. Springer, Cham. https://doi.org/10.1007/978-3-030-77980-1_14

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