On the Lagged Diffusivity Method for the Solution of Nonlinear Finite Difference Systems
<p>Summary of the nonlinear terms present at each time level and at each lagging iteration for different time discretizations. (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi>θ</mi> <mo>−</mo> </mrow> </semantics> </math>method; (<b>b</b>) IMEX scheme with explicit treatment of <math display="inline"> <semantics> <mrow> <mi mathvariant="bold-italic">G</mi> <mo>(</mo> <mi mathvariant="bold-italic">u</mi> <mo>)</mo> </mrow> </semantics> </math>; (<b>c</b>) IMEX scheme with non-constant velocity term <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">v</mi> <mo stretchy="false">˜</mo> </mover> <mrow> <mo>(</mo> <mi mathvariant="bold-italic">u</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> treated explicitly. Here <math display="inline"> <semantics> <mrow> <msub> <mi>A</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold-italic">u</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> represents the part of <span class="html-italic">A</span> dependent on <math display="inline"> <semantics> <mrow> <mi>σ</mi> <mo>(</mo> <mi mathvariant="bold-italic">u</mi> <mo>)</mo> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>A</mi> <mo stretchy="false">˜</mo> </mover> <mrow> <mo>(</mo> <mi mathvariant="bold-italic">u</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> represents the part of <span class="html-italic">A</span> dependent on <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">v</mi> <mo stretchy="false">˜</mo> </mover> <mrow> <mo>(</mo> <mi mathvariant="bold-italic">u</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math>.</p> "> Figure 2
<p>Starting vectors and tolerances of the iterative methods.</p> "> Figure 3
<p>Systems arising from the discretization of the initial PDE and from the used iterative methods.</p> "> Figure 4
<p>Blow-up solutions and representation of where and how the blow-up occurs.</p> "> Figure 5
<p>Evolution of the computed solution as <span class="html-italic">t</span> approaches <span class="html-italic">T</span> in the cases <math display="inline"> <semantics> <mrow> <msup> <mi>u</mi> <mo>*</mo> </msup> <mo>=</mo> <msubsup> <mi>u</mi> <mn>2</mn> <mo>*</mo> </msubsup> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <msup> <mi>u</mi> <mo>*</mo> </msup> <mo>=</mo> <msubsup> <mi>u</mi> <mn>4</mn> <mo>*</mo> </msubsup> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msup> <mi>u</mi> <mo>*</mo> </msup> <mo>=</mo> <msubsup> <mi>u</mi> <mn>5</mn> <mo>*</mo> </msubsup> </mrow> </semantics> </math>.</p> "> Figure 6
<p>Last computed solution for the problems in <a href="#algorithms-10-00088-f004" class="html-fig">Figure 4</a> for <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mi>t</mi> <mo>=</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics> </math>. (<b>a</b>) <math display="inline"> <semantics> <msub> <mi>u</mi> <mn>2</mn> </msub> </semantics> </math>, <math display="inline"> <semantics> <mrow> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.993</mn> </mrow> </semantics> </math>; (<b>b</b>) <math display="inline"> <semantics> <msub> <mi>u</mi> <mn>3</mn> </msub> </semantics> </math>, <math display="inline"> <semantics> <mrow> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.972</mn> </mrow> </semantics> </math>; (<b>c</b>) <math display="inline"> <semantics> <msub> <mi>u</mi> <mn>4</mn> </msub> </semantics> </math>, <math display="inline"> <semantics> <mrow> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.789</mn> </mrow> </semantics> </math>; (<b>d</b>) <math display="inline"> <semantics> <msub> <mi>u</mi> <mn>5</mn> </msub> </semantics> </math>, <math display="inline"> <semantics> <mrow> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.806</mn> </mrow> </semantics> </math>.</p> "> Figure 7
<p>Solution of the boundary layer test problem for <span class="html-italic">N</span> = 250; inner solver: BiCGstab(4).</p> ">
Abstract
:1. Introduction
- compare different discretizations. In particular, we consider what happens when the space is discretized by an upwind method or by central finite differences. We also describe different time discretizations, considering the -method and the IMEX schemes;
- study numerically what happens in some particularly critical cases, where some smoothness assumptions are not satisfied. This is the case of boundary layer and blow-up solutions.
2. Discretization
2.1. Space Discretization
- Discretization by central finite differences (error )
- Discretization by upwind scheme (error ), case > 0
- when we use the upwind scheme, the order of the approximation is ; when we use central FD, the order of the approximation is ;
- when we use the upwind scheme, is irreducibly diagonally dominant [11] (p. 23) irrespective of the choice of h. Having also positive diagonal elements and non positive off-diagonal elements, it is always a non-singular M-matrix [11] (p. 91). With central FD, is irreducibly diagonally dominant (and a non-singular M-matrix) only if h satisfies a condition on the step-size. In particular, by Equation (4) if is easy to find that this condition corresponds to
2.2. Time Discretization
3. The Lagged Diffusivity Method
3.1. Effect of the Discretization
3.2. Monotonicity of the Finite-Difference Operator and Convergence Analysis
- the functions and g are continuous in u and continuously differentiable in . Moreover, the functions and s are continuous in their variables;
- is uniformly bounded in and : there exist two positive constants and such that . Moreover, we also assume and that is constant;
- for fixed , the function is Lipschitz continuous in u (uniformly in ) with constant ;
- for fixed , the function g is uniformly monotone in u (uniformly in ) with constant [15] (p. 114); moreover, it is continuously differentiable in u.
4. Solution Procedure
4.1. Solution of the Inner Systems
4.2. Starting Vectors and Stopping Criteria
4.3. Summary of the Solution Method
Algorithm 1 Lagged diffusivity procedure | |
Require: initial condition at ; a tolerance | |
1: for do | Time level |
2: Initialize solution vector for lagged iteration: | |
3: Initialize lagged tol.: | |
4: for do | Lagged iteration |
5: Initialize linear solver tol.: | |
6: for do | Simpl. Newton iteration |
7: for do | Linear solver iteration |
8: Compute -th iterate | |
9: Compute residual of the linear system | |
10: if then break | |
11: j = j+1 | |
12: end for | |
13: Compute Newton residual | |
14: if then break | |
15: Update linear solver tol.: | |
16: k = k+1 | |
17: end for | |
18: Update vectors and matrices: find | |
19: | |
20: | |
21: if then return | |
22: end for | |
23: | |
24: end for |
5. Numerical Experiments
5.1. Effect of the Discretization Scheme
5.2. Blow-Up
5.2.1. Preliminary Results and Analysis
5.2.2. Effect of Refining the Time Grid
- the blow-up occurs on the boundary, where the solution is given by the Dirichlet boundary conditions. Thus, at inner points, where we compute the solution vector, the maximum of is always smaller than the maximum of at boundary points when ;
- as a consequence of the previous point, the smaller the number of grid points, the smaller . Indeed, we get farther from the boundary and, thus, from the singularity. This is also shown in the next subsection, where we observe that increases when we increase the number of points of the space grid;
- the singularity occurs abruptly when is reached, so the solution is still quite small also at times quite close to T. Next, we show that by further reducing , we are able to get nearer to T, leading to an increase of .
5.2.3. Effect of Refining the Space Discretization
5.3. Boundary Layer
6. Conclusions
Author Contributions
Conflicts of Interest
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Discretization | Lin. Solver | rel. | ||||||
---|---|---|---|---|---|---|---|---|
Central FD | AM | 91,812 | 7581 | |||||
BiCG(1) | 91,812 | 1199 | ||||||
Upwind | AM | 7581 | ||||||
BiCG(1) | 1199 | |||||||
Central FD | AM | 6608 | ||||||
BiCG(1) | 979 | |||||||
Upwind | AM | 6710 | ||||||
BiCG(1) | 967 | |||||||
Central FD | AM | 594 | ||||||
BiCG(1) | - | - | - | - | - | - | ||
Upwind | AM | 748 | ||||||
BiCG(1) | 1534 | |||||||
Central FD | AM | - | - | - | - | - | - | |
BiCG(1) | - | - | - | - | - | - | ||
Upwind | AM | 552 | ||||||
BiCG(1) | 1398 |
Discretization | Lin. Solver | rel. | ||||||
---|---|---|---|---|---|---|---|---|
Central FD | AM | - | - | - | - | - | - | |
BiCG(1) | - | - | - | - | - | - | ||
BiCG(2) | 807 | |||||||
BiCG(4) | 382 | |||||||
Upwind | AM | 552 | ||||||
BiCG(1) | 1398 | |||||||
BiCG(2) | 673 | |||||||
BiCG(4) | 299 | |||||||
Central FD | AM | - | - | - | - | - | - | |
BiCG(1) | - | - | - | - | - | - | ||
BiCG(2) | 629 | |||||||
BiCG(4) | 296 | |||||||
Upwind | AM | 370 | ||||||
BiCG(1) | - | - | - | - | - | - | ||
BiCG(2) | 514 | |||||||
BiCG(4) | 247 | |||||||
Central FD | AM | - | - | - | - | - | - | |
BiCG(1) | - | - | - | - | - | - | ||
BiCG(2) | 1077 | |||||||
BiCG(4) | 507 | |||||||
Upwind | AM | 322 | ||||||
BiCG(1) | - | - | - | - | - | - | ||
BiCG(2) | 476 | |||||||
BiCG(4) | 233 |
t | rel. | |||||||
---|---|---|---|---|---|---|---|---|
, | ||||||||
369 | 19 | 185 | ||||||
1209 | 21 | 202 | ||||||
25 | 254 | |||||||
, | ||||||||
592 | 20 | 186 | ||||||
1828 | 21 | 208 | ||||||
25 | 263 | |||||||
, | ||||||||
112 | 17 | 120 | ||||||
346 | 19 | 138 | ||||||
3077 | 22 | 175 |
rel. | |||||||||
---|---|---|---|---|---|---|---|---|---|
8301 |
rel. | |||||||||
---|---|---|---|---|---|---|---|---|---|
rel. | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Lin. Solver | rel. | ||||||
---|---|---|---|---|---|---|---|
BiCG(1) | 31 | 1925 | |||||
BiCG(2) | 31 | 977 | |||||
BiCG(4) | 31 | 460 | |||||
BiCG(1) | 31 | 1389 | |||||
BiCG(2) | 31 | 708 | |||||
BiCG(4) | 31 | 366 | |||||
BiCG(1) | 31 | 638 | |||||
BiCG(2) | 31 | 156 | |||||
BiCG(4) | 31 | 314 |
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Mezzadri, F.; Galligani, E. On the Lagged Diffusivity Method for the Solution of Nonlinear Finite Difference Systems. Algorithms 2017, 10, 88. https://doi.org/10.3390/a10030088
Mezzadri F, Galligani E. On the Lagged Diffusivity Method for the Solution of Nonlinear Finite Difference Systems. Algorithms. 2017; 10(3):88. https://doi.org/10.3390/a10030088
Chicago/Turabian StyleMezzadri, Francesco, and Emanuele Galligani. 2017. "On the Lagged Diffusivity Method for the Solution of Nonlinear Finite Difference Systems" Algorithms 10, no. 3: 88. https://doi.org/10.3390/a10030088