Reduced basis method applied to a convective stability problem
Abstract
Numerical reduced basis methods are instrumental to solve parameter dependent partial differential equations problems in case of many queries. Bifurcation and instability problems have these characteristics as different solutions emerge by varying a bifurcation parameter. Rayleigh–Bénard convection is an instability problem with multiple steady solutions and bifurcations by varying the Rayleigh number. In this paper the eigenvalue problem of the corresponding linear stability analysis has been solved with this method. The resulting matrices are small, the eigenvalues are easily calculated and the bifurcation points are correctly captured. Nine branches of stable and unstable solutions are obtained with this method in an interval of values of the Rayleigh number. Different basis sets are considered in each branch. The reduced basis method permits one to obtain the bifurcation diagrams with much lower computational cost.
Keywords
Reduced basis Linear stability Eigenvalues and eigenfunctions Bifurcation Rayleigh Bénard instability Convective flowAbbreviations
 RB
reduced basis
 R
Rayleigh number
 Pr
Prandtl number
 GL
Gauss–Lobatto
 Re
real part of the eigenvalue
 Im
imaginary part of the eigenvalue
 \(R_{c}\)
critical Rayleigh number
1 Introduction
Bifurcations and instabilities in differential equations are features that allow the explanation of many fluid dynamics phenomena in nature and industrial processes [1]. An example is the Rayleigh–Bénard convection problem [2, 3]. Rayleigh–Bénard and related natural convection phenomena are usual in many industrial applications. For instance, in the formation of microstructures during the cooling of molten metals in computer chips or large scale equipments. The model equations in this case are the incompressible Navier–Stokes equations coupled with a heat equation under the Boussinesq approximation. Here the conductive solution becomes unstable for a critical vertical temperature gradient beyond a certain threshold and therefore a convective motion sets in, and, depending on boundary conditions and other external physical parameters, new convective patterns occur [1].
All these problems usually need to be solved with numerical methods. To find all the different solutions for the same or different values of the bifurcation parameter and bifurcations among them specific continuation techniques are required. These techniques are highly developed for ordinary differential equations [4], but are less advanced for partial differential equations. Some continuation methods consider a perturbation with the eigenfunctions at the bifurcation point in order to find the bifurcated solution, others are based on the existence of finite dimensional inertial manifolds [5] and projections on this manifold [6], other use proper orthogonal decomposition (POD) [7].
In [8, 9] a Rayleigh–Bénard problem is studied under the perspective of looking for the bifurcation diagrams. The different solutions and successive bifurcations when the temperature gradients increase are obtained based on a branch continuation technique. In the study of these bifurcation problems the model of partial differential equations must be solved for lots of values of the bifurcation parameter and a linear stability analysis has to be performed for each solution in order to know its linear stability properties and the succession of bifurcations.
The reduced basis method is a meaningful numerical technique to solve problems of partial differential equation for a large amount of values of the bifurcation parameter with a reduced cost [10, 11, 12, 13, 14, 15, 16, 17]. This method consists of the construction of a basis of solutions for different values of this parameter. These solutions are obtained in a preliminary stage with a standard discretization and a greedy selection is applied to them. A further use of a Galerkin method for the reduced basis expansion is implemented.
In this work the reduced basis method is applied as a continuation technique to find the multiple steady solutions and instabilities among them, that appear in a Rayleigh–Bénard convection problem in a rectangle. In [18] the reduced basis method has been applied to this problem to obtain some stable solutions, allowing one to prove the efficiency of the given method to obtain solutions for different values of the parameters. The aim of the present paper is to complete the study of bifurcations with the calculation of the whole bifurcation diagram with all the solutions including the unstable ones, their linear stability, bifurcations among them and the capturing of the bifurcation points. All the steps are solved with reduced basis.
The article is organized as follows. In Sect. 2 we formulate the problem, providing the description of the physical setup, the basic equations and boundary conditions. We describe the numerical stationary problem and the linear stability analysis of the stationary solutions. Section 3 discusses the numerical reduced basis method. Section 4 describes the numerical results. Finally Sect. 5 presents the conclusions.
2 Formulation of the problem
2.1 Stationary equations
2.2 Linear stability of the stationary problem
We choose the value of the aspect ratio \(\Gamma=3.495\) as in [8, 18] and let the Rayleigh number R vary in the interval: \([1000; 3000]\). In this interval two pitchfork bifurcations and two inverse pitchfork bifurcations appear [18]. In this problem we take into account nine different solutions. In the interval \([1000;1102]\) there is only the zero solution \(S_{0}\), in the interval \([1101; 1252]\) there are two other solutions, three in total, \(S_{0}\), \(S_{1}\) and \(S_{2}\), in the interval \([1252;1538]\) five solutions \(S_{0}\)–\(S_{4}\), and finally in the interval \([1538;3000]\) nine solutions \(S_{0}\)–\(S_{8}\). Therefore solution \(S_{0}\) exists in the interval \([1000;3000]\), solutions \(S_{1}\) and \(S_{2}\) in the interval \([1102;3000]\), solutions \(S_{3}\) and \(S_{4}\) in the interval \([1252;3000]\) and solutions \(S_{5}\)–\(S_{8}\) in the interval \([1538;3000]\).
3 Numerical reduced basis method
The numerical method is based on the approximation of all the solutions by the linear combination of some well chosen solutions computed for some particular values of R, the same ones for every variable: u, θ, P. These values are obtained in a greedy fashion.
3.1 Construction of the reduced basis

First we fix the solutions we want to calculate and the interval of values of R where these solutions exist.

We solve numerically the stationary equations (6)–(8) with boundary conditions (4)–(5), with the Newton Legendre collocation method described in the previous section, for different values of the Rayleigh number R chosen on a subset of the interval, denoted as “trial set” \(\Xi_{\mathrm{trial}}\). Some values of the Rayleigh number are equidistant, and others are near the bifurcation points. We name the associated solutions \(\Phi(R) \equiv( \textbf {u}(R), \theta(R), P(R))\).
 For the first step \(i=1\), we choose a value of the Rayleigh number that we name \(R_{1}\), with its corresponding solution \(\Phi(R_{1})\), i.e. in this work it is the smallest value of R in the interval we consider. We normalize this stationary solution according to the \(L^{2}\) scalar product:then we consider a first space \(X_{1}=\mbox{span}\{ \psi^{\textbf{u}}_{1}\}\times\mbox{span}\{ \psi^{\theta}_{1}\} \times\mbox{span}\{ \psi^{P}_{1}\} \).$$\Psi_{1}= \biggl( \psi_{1}^{\textbf{u}} = \frac{\textbf{u}_{1}}{\ \textbf{u}_{1}\_{L^{2}}}, \psi_{1}^{\theta}= \frac{{\theta}_{1}}{\ \theta_{1} \_{L^{2}}}, \psi_{1}^{P} = \frac {{P}_{1}}{\ P_{1} \_{L^{2}}} \biggr), $$

We introduce the projection operator \(\Pi_{X_{1}^{\textbf{u}}} \times\Pi_{X_{1}^{\theta}} \times\Pi_{X_{1}^{P}}\) onto \(X_{1}\) for the \(L^{2}\) inner product and consider the approximation \((\textbf{u}^{(1)}(R), \theta^{(1)}(R), P^{(1)}(R)) = [\Pi_{X_{1}^{\textbf {u}}} (\textbf{u}(R)),\Pi_{X_{1}^{\theta}} (\theta(R)), \Pi _{X_{1}^{P}}(P(R))]\) for every R. Note that it corresponds to the product of independent projection operators \(\Pi_{X_{1}^{\textbf{u}}}\), \(\Pi_{X_{1}^{\theta}}\) and \(\Pi_{X_{1}^{P}}\).
 We evaluate the relative errors of the projections on \(X_{1}\) for the velocity and temperature fields u, θ on one side and the pressure P on the other side:and$$\epsilon^{(1)}_{1}(R) = \frac{\ (u_{x}(R), u_{z}(R),\theta (R))(u^{(1)}_{x}(R),u^{(1)}_{z}(R),\theta^{(1)}(R))\_{(L^{2})^{3}}}{\ (u_{x}(R), u_{z}(R),\theta(R))\_{(L^{2})^{3}}} $$over any values of R chosen on \(\Xi_{\mathrm{trial}}\). We then choose \(R_{2}\) as follows:$$\epsilon^{(1)}_{2}(R)= \frac{\ P(R)P^{(1)}(R)\_{L^{2}}}{\P(R)\_{L^{2}}}, $$and the corresponding stationary solution is \(\Phi(R_{2})\).$$R_{2} = \underset{R\in\Xi_{\mathrm{trial}}}{\operatorname{argmax}} \max_{j=1, 2} \epsilon^{(1)}_{j}(R) $$

Given step i, we orthonormalize the \(i+1\) functions by Gram–Schmidt procedure and we consider the \(i+1\) space \(X_{i+1} = \mbox{span}\{ \psi^{\textbf{u}}_{1}, \ldots,\psi^{\textbf{u}}_{i+1}\}\times\mbox{span}\{ \psi^{\theta}_{1}, \ldots,\psi^{\theta}_{i+1}\} \times\mbox{span}\{ \psi^{P}_{1}, \ldots,\psi^{P}_{i+1}\}\).
 We introduce the projection operator \(\Pi_{X_{i+1}^{\textbf{u}}} \times\Pi_{X_{i+1}^{\theta}} \times\Pi_{X_{i+1}^{P}}\) onto \(X_{i+1}\) for the \(L^{2}\) inner product and consider the approximationfor every R.$$\bigl(\textbf{u}^{(i+1)}(R), \theta^{(i+1)}(R), P^{(i+1)}(R)\bigr) =\bigl[\Pi _{X_{i+1}^{\textbf{u}}} \bigl(\textbf{u}(R)\bigr), \Pi_{X_{i+1}^{\theta}} \bigl(\theta (R)\bigr), \Pi_{X_{i+1}^{P}}\bigl(P(R)\bigr) \bigr] $$
 Again, we evaluate the relative errors of the projections on \(X_{i+1}\) for the velocity and temperature fields u, θ on one side and the pressure P on the other side:and$$\epsilon^{(i+1)}_{1}(R) = \frac{\ (u_{x}(R), u_{z}(R),\theta (R))(u^{(i+1)}_{x}(R),u^{(i+1)}_{z}(R),\theta^{(i+1)}(R))\ _{(L^{2})^{3}}}{\ (u_{x}(R), u_{z}(R),\theta(R))\_{(L^{2})^{3}}} $$over any values for R chosen on \(\Xi_{\mathrm{trial}}\). We then choose \(R_{i+2}\) as follows:$$\epsilon^{(i+1)}_{2}(R)=\frac{\P(R)P^{(i+1)}(R)\_{L^{2}}}{\P(R)\_{L^{2}}}, $$and the corresponding stationary solution is \(\Phi(R_{i+2})\).$$R_{i+2} = \underset{R\in\Xi_{\mathrm{trial}}}{\operatorname{argmax}} \max_{j=1, 2} \epsilon^{(i+1)}_{j}(R) $$

This procedure is repeated until we reach a value \(N < \hbox{card}(\Xi_{\mathrm{trial}})\) for which the stopping criterium \(\epsilon^{(N)}_{j} \leq10^{7}\), \(j=1,2\) is satisfied.
Number of snapshots in the trial set for the different reduced basis (RB) for different solution
\(S_{1}\)  \(S_{3}\)  \(S_{3}\) (two sets)  \(S_{5}\)  \(S_{5}\) (three sets)  

# \(\Xi_{\mathrm{trial}}\)  23  22  29, 19  19  22, 26, 51 
# RB  9  8  6, 6  10  6, 7, 7 
\(\epsilon^{(j)}_{1}\), \(\epsilon^{(j)}_{2}\), \(j=1,\ldots,N\) and the respective Rayleigh number R in which the maximum takes place for different dimensions j of the reduced basis space. R is in the interval \([1102;3000]\) for solutions \(S_{1}\)
j  \(\epsilon^{(j)}_{1}\)  \(\epsilon^{(j)}_{2} \)  R 

1  0.809  0.236  1102 
2  0.085  0.0.009  3000 
3  0.008  0.001  1500 
4  0.003  4.7⋅10^{−4}  2200 
5  3.3⋅10^{−4}  6.7⋅10^{−5}  1110 
6  1.1⋅10^{−4}  1.7⋅10^{−5}  2700 
7  1.7⋅10^{−5}  2.7⋅10^{−6}  1900 
8  2.8⋅10^{−6}  4.1⋅10^{−7}  1800 
9  1.8⋅10^{−7}  1.7⋅10^{−8}  1300 
\(\epsilon^{(j)}_{1}\), \(\epsilon^{(j)}_{2}\), \(j=1,\ldots,N\) and the respective Rayleigh number R in which the maximum takes place for different dimensions j of the reduced basis space. R is in the interval \([1253;3000]\) for solutions \(S_{3}\)
j  \(\epsilon^{(j)}_{1}\)  \(\epsilon^{(j)}_{2} \)  R 

1  0.748  0.588  1253 
2  0.056  0.015  3000 
3  0.007  0.001  1600 
4  0.002  2.5⋅10^{−4}  2200 
5  1.1⋅10^{−4}  3.5⋅10^{−5}  1300 
6  2.6⋅10^{−5}  7.4⋅10^{−6}  2700 
7  4.6⋅10^{−6}  7.5⋅10^{−7}  1400 
8  7.0⋅10^{−7}  9.6⋅10^{−8}  1260 
\(\epsilon^{(j)}_{1}\), \(\epsilon^{(j)}_{2}\), \(j=1,\ldots,N\) and the respective Rayleigh number R in which the maximum takes place for different dimensions j of the reduced basis space. R is in the interval \([1253;1538]\) for solutions \(S_{3}\) in the upper part of the table and R is in the interval \([1538;3000]\) for solutions \(S_{3}\) in the lower part of the table
j  \(\epsilon^{(j)}_{1}\)  \(\epsilon^{(j)}_{2} \)  R 

1  0.354  0.198  1253 
2  0.011  0.002  1530 
3  6.4⋅10^{−4}  1.2⋅10^{−4}  1330 
4  6.5⋅10^{−5}  1.2⋅10^{−5}  1270 
5  1.4⋅10^{−6}  2.9⋅10^{−7}  1450 
6  2.7⋅10^{−7}  5.2⋅10^{−8}  1380 
1  0.442  0.412  1539 
2  0.018  0.005  3000 
3  0.001  2.7⋅10^{−4}  2100 
4  2.4⋅10^{−4}  5.2⋅10^{−5}  1700 
5  5.9⋅10^{−6}  8.9⋅10^{−7}  2600 
6  5.5⋅10^{−7}  2.1⋅10^{−7}  1900 
\(\epsilon^{(j)}_{1}\), \(\epsilon^{(j)}_{2}\), \(j=1,\ldots,N\) and the respective Rayleigh number R in which the maximum takes place for different dimensions j of the reduced basis space. R is in the interval \([1538;3000]\) for solutions \(S_{5}\)
j  \(\epsilon^{(j)}_{1}\)  \(\epsilon^{(j)}_{2} \)  R 

1  2.365  1.000  1539 
2  0.290  0.162  3000 
3  0.035  0.010  1800 
4  0.010  0.003  2300 
5  0.001  4.7⋅10^{−4}  1580 
6  3.9⋅10^{−4}  7.2⋅10^{−5}  2700 
7  1.1⋅10^{−4}  3.1⋅10^{−5}  1550 
8  1.6⋅10^{−5}  4.4⋅10^{−6}  2000 
9  2.3⋅10^{−6}  8.6⋅10^{−7}  1700 
10  3.9⋅10^{−7}  9.4⋅10^{−8}  2900 
\(\epsilon^{(j)}_{1}\), \(\epsilon^{(j)}_{2}\), \(j=1,\ldots,N\) and the respective Rayleigh number R in which the maximum takes place for different dimensions j of the reduced basis space. R is in the interval \([1539;1600]\) for solutions \(S_{5}\) in the first part of the table, R is in the interval \([1600;2000]\) for solutions \(S_{5}\) in the second part of the table and R is in the interval \([2000;3000]\) for solutions \(S_{5}\) in the last part of the table
j  \(\epsilon^{(j)}_{1}\)  \(\epsilon^{(j)}_{2} \)  R 

1  0.785  0.265  1539 
2  0.016  0.006  1600 
3  3.1⋅10^{−4}  8.5⋅10^{−4}  1560 
4  4.4⋅10^{−5}  1.1⋅10^{−5}  1545 
5  1.1⋅10^{−6}  3.0⋅10^{−7}  1585 
6  7.2 ⋅10^{−7}  2.0⋅10^{−8}  1541 
1  1.154  0.683  1600 
2  0.050  0.026  2000 
3  0.003  6.6⋅10^{−4}  1740 
4  3.5⋅10^{−4}  1.2⋅10^{−4}  1880 
5  1.3⋅10^{−5}  7.8⋅10^{−6}  1640 
6  2.7⋅10^{−6}  6.4⋅10^{−7}  1940 
7  2.4⋅10^{−7}  5.0⋅10^{−8}  1610 
1  0.834  0.554  2000 
2  0.039  0.022  3000 
3  0.002  7.0⋅10^{−4}  2380 
4  3.4⋅10^{−4}  8.9⋅10^{−5}  2700 
5  1.2⋅10^{−5}  4.0⋅10^{−6}  2120 
6  2.4⋅10^{−6}  9.2⋅10^{−7}  2880 
7  1.3⋅10^{−7}  4.6⋅10^{−8}  2040 
3.2 Galerkin procedure
Note that the construction of the matrix \(M^{s}\) can be done online very efficiently in \(\mathcal{O}(N^{3})\) operations if, during the preprocessing offline stage, double integrals involving the elements of the reduced basis are computed, we are indeed in the case where the appearance of the parameter is outside of the integrals and the problem is only slightly nonlinear (bilinear), during the offline stage the integrals are calculated using the Legendre Gauss–Lobatto quadrature formulas [24].
3.3 Linear stability analysis
3.4 Postprocessing
The reduced basis is formed by functions that are not solutions of the Galerkin procedure, indeed these are solutions of a Legendre collocation method. We introduce a rectification postprocessing presented in [18], which consists of a change of basis from the reduced basis to the Galerkin solutions on the values of R of the basis as is explained as follows:
We start by computing the reduced basis Galerkin approximations for all values \(R=R_{i}\), \(i=1,\dots,N\), that are used in the reduced basis construction. This gives us coefficients \(\textbf{u}_{N}(R_{i}) = \sum^{N}_{j=1} \alpha^{i}_{j} \psi^{\textbf{u}}_{j}\) and \({ {\theta}_{N} (R_{i})}= \sum^{N}_{j=1} \beta^{i}_{j} \psi^{\theta}_{j}\). We name \(Q^{u}\) (resp. \(Q^{\theta} \)) the matrix with entries equal to \(\alpha^{i}_{j}\) (resp. \(\beta^{i}_{j}\)). We call \(S^{u}\) (resp. \(S^{\theta}\)) the matrix with columns equal to the coordinates of \(\textbf{u}(R_{i})\) (resp. \(\theta(R_{i})\)) in the reduced basis \(\psi^{\textbf{u}}_{j}\) (resp. \(\psi^{\theta}_{j}\)), \(j=1,\dots,N\). Finally, we set \(P^{u}=S^{u} [Q^{u}]^{1}\) (resp. \(P^{\theta}=S^{\theta}[Q^{\theta} ]^{1}\)). This part is done during the offline stage and the matrix is stored.
4 Results and discussion
4.1 Linear stability analysis
Figure 3(b) shows the norm of the difference between the real part of the eigenvalue with largest real part obtained via Legendre collocation and by a postprocessed reduced basis method based on Legendre collocation for solutions \(S_{3}\) in the interval of R\([1534; 1539]\) near the bifurcation point at \(R_{c3}=1538\). This difference is \({\mathcal {O}}(10^{4})\) (where both approximations cross near the bifurcation point), increasing as we move away from the bifurcation point. Figure 3(c) shows the norm of the difference between the real part of the eigenvalue with largest real part obtained via Legendre collocation and by a postprocessed reduced basis method based on Legendre collocation divided by the norm of the real part of the eigenvalue obtained with Legendre collocation for solutions \(S_{3}\) in the interval of R\([1534; 1539]\) near the bifurcation point at \(R_{c3}=1538\). This difference is \({\mathcal {O}}(10^{1})\) except around \(R=1537\) where it is \(O(1)\) because the real part of the eigenvalue crosses the axis and becomes zero near this point at \(R_{c3\mathrm{L}}=1537.0860\), for this reason the relative error becomes maximum at this value \(R=1537\).
4.2 Capturing the bifurcation points
The point where different types of solutions intersect or get together is going to be the bifurcation point, where the bifurcation takes place. The first pitchfork bifurcation takes place at \(R=1101\), where solutions \(S_{1}\) and \(S_{2}\) get together. The second bifurcation occurs at \(R=1252\), where solutions \(S_{3}\) and \(S_{4}\) intersect. Finally both secondary inverse pitchfork bifurcations take place at \(R=1538\), where \(S_{5}\) and \(S_{6}\) intersect, and where \(S_{7}\) and \(S_{8}\) get together. These bifurcation points are correctly captured with the reduced basis method by plotting the solutions in the bifurcation diagram in Fig. 4. The bifurcations are pitchfork because symmetric solutions appear at those points.
The bifurcation points are also captured with the linear stability analysis as explained in Sect. 4.1.
4.3 Errors on the solutions
4.4 Advantages of reduced basis method
The reduced basis method is supported by standard discretizations. The offline work for the calculation of the solutions to construct the reduced basis needs these standard methods. But, once the work of the standard method is done, the use of the reduced basis has several advantages.
The pressure variable disappears in the Galerkin approach due to the variational formulation and the incompressibility of the fluid (\(\nabla \cdot\vec{v}=0\)). This fact, together with the few modes required for the Galerkin expansion, provides small matrices with the reduced bases discretization. For a single value of the Rayleigh number R the size of the matrices that appear after the discretizations is 2016 in Legendre collocation with expansions of order \(13 \times35\), whereas in the case of the reduced basis with 8 elements the size of matrices is 16. A factor of 126 in the size of the matrices for each value of R. Legendre collocation matrices are dense by diagonal blocks of size \(14 \times36\).
The behavior of the Newton method for the nonlinearity is improved with respect to standard methods. A branch of solutions of a type refers to the solutions of that kind for different values of the parameter R in the interval where they exist. In the Legendre collocation method case to calculate solutions in a new branch a continuation technique is required, in this case it is based in adding the eigenfunction of the linear stability analysis near the bifurcation point to the base solution as initial guess in the Newton method. Once a solution in a branch is obtained to reach the solution in this branch for a larger value of the Rayleigh number the new solution is considered as initial guess. For a large value of the Rayleigh number the solution must be calculated by increasing slowly the Rayleigh number. For instance, we obtain the first solution in the branch in the interval \([1101; 3000]\) near \(R=1101\). To obtain the solution at \(R=3000\) we need to calculate the solution at \(R=1102\), take this solution as initial guess for \(R=1110\) and calculate the solutions increasing the value of R in steps of 10 till \(R=3000\). Sometimes the steps of increase on R can be larger. So, it is not possible to jump from \(R=1101\) till \(R=3000\) with Legendre collocation. In the reduced basis this is not the case, the solution can be directly calculated for any value of R. The reason for this behavior must be that nothing drive the solutions to be attracted by a different branch since there is not unexpected elements in the basis set. Also solutions obtained with reduced basis method are a great help as guess solutions for the Newton method in Legendre collocation. In fact the Legendre solutions necessary to valuate the errors have been calculated solving first with Galerkin reduced basis and taking this solution as initial guess for Legendre.
The number of operations is drastically reduced. If we calculate the branch of solutions \(S_{5}\) taking steps of 10 in R in the interval \([1538; 3000]\), 146 values of R are required, for the branch in the interval \([1101; 3000]\), 190 values of R and for the branch in the interval \([1252; 3000]\) 175 values of R. Therefore the whole diagram requires 1314 values of R. If we take into account symmetries only 511 values of R are necessary. The problem is nonlinear, if we consider an average of 10 iterations for each problem and we solve the linear systems with a method with \({\mathcal {O}}(N^{2})\) operations, being N the size of the matrices, for each value of R we solve the system with \({\mathcal {O}}(10^{7})\) operations for Legendre collocation and with \({\mathcal {O}}(10^{3})\) with reduced basis. Then multiplying by 10^{3} values of R, \({\mathcal {O}}(10^{10})\) operations for Legendre collocation and \(O(10^{6})\) for reduced basis to obtain the whole bifurcation diagram. The offline number of operations of reduced basis requires to solve Legendre for order 10 values of the Rayleigh number, therefore \({\mathcal {O}}(10^{8})\) operations.
Summarizing, for a single value of the parameter R, the offline number of operations is \({\mathcal {O}}(10^{8})\), the online maximum number of operations for Legendre collocation is \({\mathcal {O}}(10^{9})\) and for reduced basis \({\mathcal {O}}(10^{3})\). For the whole bifurcation diagram the offline number of operations is \({\mathcal {O}}(10^{8})\), the online number of operations for Legendre collocation is \({\mathcal {O}}(10^{10})\) and for reduced basis \({\mathcal {O}}(10^{6})\).
A significant advantage of solving the eigenvalue problem is provided by the use of a reduced basis, which is due to a large reduction in computational cost. This reduction arises due to the size of the matrices in the eigenvalue problem; in Legendre collocation the size of the matrix is 2016, while using a reduced basis with 8 elements it is only 16. The eigenvalue problem is solved with an adaptation of the implicitly restarted Lanczos method [25]. This method has computational complexity \(O(N^{3})\), where N is the size of the matrix. Therefore for a fixed value of R in Legendre collocation the complexity is \(O(10^{9})\) whereas for reduced basis it is only \(O(10^{3})\), six orders lower. For 1000 values of the Rayleigh number Legendre collocation reaches \(O(10^{12})\) and reduced basis \(O(10^{6})\). This is reflected in the temporal computational cost, which is 122 s when using Legendre collocation and only it is 6 s using a reduced basis. Therefore the reduction is of a factor of 20 in time.
5 Conclusions
We have solved a Rayleigh–Bénard problem in a rectangle using a reduced basis method in an interval of values of the Rayleigh number R where there are four bifurcations and nine different solutions. Different basis have been considered in each branch of solutions. A linear stability analysis on those solutions has been performed with reduced basis. The bifurcation points are correctly captured with the reduced basis method looking at the intersection of branches of solutions or regarding the eigenvalues of the linear stability analysis. The relative errors on the solutions in the unstable branches are of the same order as the stable ones, \({\mathcal {O}}(10^{2})\), and \({\mathcal {O}}(10^{4})\) after a postprocessing, and become worse near the bifurcation points. Near those points the errors are divided by the norm of solutions that are disappearing, for this reason relative errors become \({\mathcal {O}}(10^{1})\). The behavior of the Newton method for the nonlinearity is improved with reduced basis method with respect to standard methods. Matrices are small with the reduced basis discretization. For a single value of the Rayleigh number R the size of the matrices that appear after the discretizations are 2016 in Legendre collocation, whereas in the case of the reduced basis with 8 elements the size of matrices are 16 because pressure disappears in this formulation. A factor of 126 in the size of the matrices for each value of R. The computational complexity is less with reduced basis method. For a single value of the parameter R, the offline number of operations is \({\mathcal {O}}(10^{8})\), the online number of operations for Legendre collocation is \({\mathcal {O}}(10^{9})\) and for reduced basis \({\mathcal {O}}(10^{3})\). For the whole bifurcation diagram the offline number of operations is \({\mathcal {O}}(10^{8})\), the online number of operations for Legendre collocation is \({\mathcal {O}}(10^{10})\) and for reduced basis \({\mathcal {O}}(10^{6})\). The advantage of solving the eigenvalue problem with reduced basis is huge as regards with respect to the large reduction of the computational cost. For a fixed value of R in Legendre collocation the complexity is \(O(10^{9})\) whereas for reduced basis it is only \(O(10^{3})\), six orders lower. This is reflected in the reduction in the computational cost in time of a factor of 20. There is a startup work in order to calculate the reduced basis, but once this is done, the reduced basis method permits to speed up the computations of these bifurcation diagrams. A study of the eigenfunction spaces obtained with reduced basis can be also of interest and it will be addressed in future work.
Notes
Acknowledgements
Not applicable.
Availability of data and materials
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Authors’ contributions
All authors contributed equally and were involved in writing the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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