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... related with the order of the partial differential equation, i.e. the order of the highest spatial derivative. A systematic way to estimate this, without having the macroscopic equation, is given in =-=[15]-=-. 3 Model homogenization problem Here, we review some basic results from homogenization theory. We note that we are interested in finding the effective behaviour of the solution. In our setup, we know...

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...hed, continuum numerical analysis tools. Recently, a so-called “equation-free” approach to the study of the coarse-grained behavior of such problems has been proposed which circumvents the first step =-=[4, 5, 6]-=-. This computational approach is based on the “coarse” or macroscopic time-stepper, a map from the coarse variables at time t = 0 to those at time t = T, where T is typically much larger than characte...

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...ons, such as no-flux or Dirichlet. We also assume that the order d of the unavailable macroscopic equation (the highest spatial derivative) is known. A strategy to obtain this information is given in =-=[22]-=-. So, we know that the macroscopic equation is of the form ∂tU = F(U,∂xU,...,∂ d xU,t), (11) where ∂t denotes the time derivative and ∂ k x denotes the k-th spatial derivative. 3.1 The gap-tooth schem...

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...t posed is a simplified version of problems arising in these fields. Recently, a number of algorithms have been developed to deal with such problems. They come under the title of “equation-free” (see =-=[1]-=- and references within). The main idea is to use short simulations of the microscale model (1) in order to locally estimate the macroscale model (3). Applying this idea to the example described above,...

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...ion is of the form ∂tU = F (U, ∂xU, . . . , ∂dxU, t), (19) in which the order of the equation (the highest spatial derivative) d is assumed to be known. For a strategy to obtain such information, see =-=[24]-=-. Suppose we want to obtain the solution of (19) on the interval [0, 1], using an equidistant, macroscopic mesh Π(∆x) := {0 = x0 < x1 = x0 + ∆x < . . . < xN = 1}. Given equation (19), we can define a ...

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...times to know the order of the highest order derivatives that appear in the effective macroscale equation, even if we do not know all the details of the macroscale model. An algorithm was proposed in =-=[19]-=- for this purpose. The “baby-bathwater scheme”, as it was called, promises to find the highest order derivative in the effective macroscale model, by performing simulations using the microscopic model...

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...g0(L(x))). Thus, the explicit approach (4.1) converges for ε→ 0 and εt→ 0 only ifR◦g0◦L equals the identity on Rd. Often this requirement is approximated by R ◦ L = I because g0 is in general unknown =-=[21, 26, 27, 33, 36]-=-. Note that there is no ε or t dependence in the limiting map R ◦ g0 ◦ L, resulting in a much more restrictive convergence condition for equation-free analysis based on the explicit map Φ(t;x) = R(Mε(...

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...der of the effective evolution equation. The design of computational experiments to determine the spatial order of an unknown (in closed form) equation is an interesting subject, discussed in part in =-=[27]-=-.sInterval S(x) 0 R. Erban et al. / Physica D 215 (2006) 1–24 9 Fig. 4. (a) Graph of “hat-profile” signal function S(x). (b) Graph of S ′ (x). (c) Graph of S ′′ (x). � 0, 1 � � 15 5 , 2 � � 25 5 , 3 �...