Convergence analysis of the fast sweeping method for static convex Hamilton–Jacobi equations
 Songting Luo^{1} and
 Hongkai Zhao^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s4068701600838
© The Author(s) 2016
Received: 10 April 2016
Accepted: 29 August 2016
Published: 14 December 2016
Abstract
In this work, we study the convergence of an efficient iterative method, the fast sweeping method (FSM), for numerically solving static convex Hamilton–Jacobi equations. First, we show the convergence of the FSM on arbitrary meshes. Then we illustrate that the combination of a contraction property of monotone upwind schemes with proper orderings can provide fast convergence for iterative methods. We show that this mechanism produces different behavior from that for elliptic problems as the mesh is refined. An equivalence between the local solver of the FSM and the Hopf formula under linear approximation is also proved. Numerical examples are presented to verify our analysis.
Keywords
Mathematics Subject Classification
1 Background
Efficient and robust iterative methods are highly desirable for solving a variety of static hyperbolic partial differential equations (PDEs) numerically. The most important property for hyperbolic problems is that the information propagates along the characteristics. For linear hyperbolic problems, the characteristics are known á priori and do not intersect. For nonlinear hyperbolic problems, the characteristics are not known á priori and may intersect. Consequently the information that propagates along different characteristics has to compromise in certain ways when the characteristics intersect to render the desired weak solution. When discretizing hyperbolic PDEs, monotone upwind schemes are an important class of schemes that use stencils following the characteristics in the direction from which the information comes (e.g., see [39]). How to design an effective iterative method for hyperbolic problems also needs to fully utilize the above properties. We use the fast sweeping method (FSM) [43] as an example to show that the combination of monotone upwind schemes with Gauss–Seidel iterations and proper orderings can provide fast convergence. The convergence mechanism is different from that of iterative methods for elliptic problems, where relaxation is the underlying mechanism for convergence and the key point is how to deal with longrange interactions through shortrange interactions efficiently by using techniques such as multigrids and/or effective preconditioners. We present both analysis and examples to explain this different convergence behavior as the mesh is refined.
 (A1)
Continuity: \(H\in C({\overline{\Omega }}\times R^d)\),
 (A2)
Convexity: \(H(\mathbf x ,\mathbf p )\) is convex in \(\mathbf p \in R^d\) for all \(\mathbf x \in {\overline{\Omega }}\),
 (A3)
Coercivity: \(H(\mathbf x ,\mathbf p ) \rightarrow \infty \) as \(\mathbf p \rightarrow \infty \) uniformly for \(\mathbf x \in {\overline{\Omega }}\),
 (A4)
Compatibility of Hamiltonian: \(H(\mathbf x ,0)\le 0\) for all \(\mathbf x \in \Omega \),
 (A5)Compatibility of Dirichlet data: \(g(\mathbf x )g(\mathbf y )\le l(\mathbf x ,\mathbf y )\) for all \(\mathbf x ,\mathbf y \in \partial \Omega \), where \(l(\mathbf x ,\mathbf y )\) is the optical distance defined by [7, 22],$$\begin{aligned} \begin{array}{l} l(\mathbf x ,\mathbf y )=\inf \left\{ \int _0^1\rho (\xi (t),\xi '(t))\mathrm{{d}}t : \xi \in C^{0,1}([0,1],{\overline{\Omega }}), \xi (0)=\mathbf x , \xi (1)=\mathbf y \right\} , \\ \\ \text{ with }~\rho (\mathbf x ,\mathbf q )=\max \limits _{H(\mathbf x ,\mathbf p )=0}\langle \mathbf p , \mathbf q \rangle \text{ the } \text{ support } \text{ function. } \end{array} \end{aligned}$$(1.2)
 (A6)
g is Lipschitz continuous.

An appropriate upwind discretization scheme (or local solver) that is consistent with the underlying PDE and guarantees the numerical solution converges to the desired weak solution.

Gauss–Seidel iterations with enforced causality: combined with an appropriate upwind scheme it means that (1) the information propagates along the characteristics efficiently, and (2) all newly updated information is used in a correct way and the intersection of different characteristics can be resolved.

Alternating orderings that can cover the propagation of the information in all directions in a systematic and efficient way.
In this paper we first explain the monotonicity and consistency of the FSM and show that there is a contraction property. Then we prove the convergence that is implied by the contraction property. We prove the convergence in two ways: one is following the usual convergence proof as in [5] by monotonicity and consistency; the other one is through its equivalence to the discretized Hopf formula in the framework of linear approximation. More importantly, we show that the contraction property implies local truncation error estimate. Furthermore, we study the contraction property of the FSM for hyperbolic problems and use it to explain the fast convergence. Through the contraction property we study the convergence of Gauss–Seidel iterations with proper orderings. By using an example with periodic boundary conditions (see examples in Sect. 4), we analyze and explain a phenomenon for hyperbolic problems which is different from that for elliptic problems: fewer iterations are needed as the mesh is refined. Since periodic boundary conditions are imposed, the discretized system cannot be put into a triangular system due to the cyclic dependence and the argument based on the method of characteristics cannot be used.
The paper is organized as follows. In Sect. 2 we recall monotone upwind schemes and the fast sweeping method. In Sect. 3 we first show the contraction property of monotone upwind schemes; then, we prove the convergence of the fast sweeping method. In Sect. 4 we show some numerical examples to verify our analysis. Conclusion remarks are given at the end.
2 Monotone upwind schemes
Remark 2.1

The time step can still be relaxed by a factor as explained above and in Fig. 1. Using an efficient implicit solver, such as the fast sweeping method, one can still gain computational efficiency.

The constant in the CFL condition for explicit schemes may not be easy to estimate for nonlinear problems. For example, when the Hamiltonian is nonlinear, the flux velocity depends on the solution itself and varies significantly such that a sharp global bound of the velocity is unavailable. This usually leads to either inefficiency or instability.
The consistency of a numerical scheme is usually easy to satisfy. However, it does not guarantee the stability and convergence of the numerical solution. For hyperbolic PDEs, an important class of discretization schemes is monotone schemes. The monotonicity is equivalent to requiring that \(F_i\) is nondecreasing in its first variable and nonincreasing in the remaining variables or vice versa. For nonlinear hyperbolic problems the monotonicity is usually necessary for the convergence of the numerical solutions to the correct weak solutions [5, 12]. Although monotone schemes are at most firstorder accurate (e.g., see [32, 39]), they provide the most robust schemes in practice as well as the starting point for highorder schemes [6, 33, 38, 42]. The simplest way to construct monotone schemes is using the Lax–Friedrichs scheme which uses central differences to approximate the derivatives and explicit numerical viscosity term with a coefficient proportional to the grid size, e.g., the discrete Laplacian, to achieve the monotonicity. The main advantage of the Lax–Friedrichs scheme is its simplicity and generality. However, it is not upwind which makes iterative methods converge slowly because the underlying equation has changed from a hyperbolic problem to an elliptic problem with singular perturbation. The convergence to the correct viscosity solution is through the vanishing viscosity as the grid size decreases. Therefore we focus on upwind monotone schemes that are more desirable for iterative methods. In particular, we study the FSM on a general mesh [34, 35, 43]. For simplicity we restrict our discussion in 2D. Extension to higher dimension is straightforward.
 1.
There are no solutions for \(u(\mathbf C )\) from (2.7), i.e., the triangle does not support any consistent candidate for \(u(\mathbf C )\), e.g., Fig. 3a.
 2.
There is only one solution for \(u(\mathbf C )\) from (2.7), i.e., the triangle supports one consistent candidate for \(u(\mathbf C )\), e.g., Fig. 3b.
 3.
There are two solutions for \(u(\mathbf C )\) from (2.7), i.e., the triangle supports two consistent candidates for \(u(\mathbf C )\), e.g., Fig. 3c.
The discretization at each vertex results in a nonlinear equation like (2.7). Since we have a boundary value problem, all the vertices are coupled together and a large system of nonlinear equations needs to be solved, which is the second step in designing the method. The key idea behind the FSM is using causality enforced Gauss–Seidel iterations with alternating orderings.
 1.
Initial guess:
For vertices on or near the boundary, their values are set according to the given boundary conditions, and they are fixed during iterations. All other vertices are assigned a large value, e.g., infinity.
 2.Causality enforced Gauss–Seidel iterations with alternating orderings (sweepings):

Update during each iteration: at a vertex \(\mathbf C \), the updated value \(u^\mathrm{{new}}(\mathbf C )\) at \(\mathbf C \) iswhere \(u^\mathrm{{old}}(\mathbf C )\) is the current value at \(\mathbf C \) and \({\tilde{u}}(\mathbf C )\) is the value at \(\mathbf C \) computed from the current given neighboring values according to (2.7) and (2.11).$$\begin{aligned} u^\mathrm{{new}}(\mathbf C )=\min \{u^\mathrm{{old}}(\mathbf C ), {\tilde{u}}(\mathbf C )\}, \end{aligned}$$(2.12)

Orderings: all vertices are ordered according to a finite number of different orderings alternately during Gauss–Seidel iterations. The orderings are designed in a way to cover all the directions of the characteristics systematically and efficiently. For example, in 2D cases, if a Cartesian grid is used, four alternating orderings are needed and given byIf a triangulated mesh is used, we can choose the orderings designed in [34, 35], where the distances to a few fixed reference points are used to order all the vertices.$$\begin{aligned} \begin{array}{l} (1)~ i=1{:}I; j=1{:}J; \qquad (2)~ i=I{:}1; j=1{:}J;\\ (3)~ i=I{:}1; j=J{:}1; \qquad (4)~ i=1{:}I; j=J{:}1. \end{array} \end{aligned}$$(2.13)

3 Properties of FSM and its convergence
In this section, we analyze the fast convergence of the FSM and the convergence of its numerical solution to the viscosity solution as the grid size approaches zero. Firstly, we prove the monotonicity and consistency, especially we prove the scheme has a contraction property. Secondly, we show the convergence of the FSM and the convergence of its numerical solution to the viscosity solution as the grid size approaches zero. We also prove its equivalence to the discretized Hopf formula. Finally, we analyze the effect of the contraction property that contracts the local truncation error, which also implies the fast convergence with optimal complexity.
3.1 Monotonicity, consistency and contraction property
We show that the scheme is consistent and monotone, and especially it has a contraction property.
Lemma 3.1
Proof
Lemma 3.2
Proof
Remark 3.3
Tsitsiklis [41] proved the same result on a Cartesian grid.
As a consequence of the monotonicity of the scheme and (3.1), we have the following lemma.
Lemma 3.4
The fast sweeping algorithm is monotone in the initial data, i.e., if at iteration k, \(u^{(k)} \le v^{(k)}\) at every grid point, then at any later iteration, say \(n>k\), \(u^{(n)} \le v^{(n)}\) at every grid point.
With the above properties for the discretization scheme, we prove the convergence of the iterative method and the convergence of the numerical solution to the viscosity solution as the grid size approaches zero. Besides, we prove the equivalence between the FSM discretization and the Hopf formula under linear approximations.
3.2 Convergence of the FSM
We first show the convergence of the FSM, which is implied by the contraction property and monotonicity. Here we use subscript h to indicate the numerical solutions.
Theorem 3.5
Proof
Remark 3.6
Now we prove that the numerical solution of the FSM converges to the viscosity solution as the grid size approaches zero.
Theorem 3.7
As \(h\rightarrow 0\), the numerical solution \(\{u_h\}\) of the FSM converges uniformly to the viscosity solution of (1.1).
Proof
The proof consists of two steps: (1) \(\{u_h\}\) is equicontinuous and uniformly bounded (Remark 3.6) such that by the theorem of Arzelà–Ascoli \(\{u_h\}\) converges to some function u satisfying (1.1) as \(h\rightarrow 0\) (e.g., see [7]); (2) u is the viscosity solution by the monotonicity (e.g., see [5]).
We first prove \(\{u_h\}\) is Lipschitz continuous by the following three steps: (a) for two vertices \(\mathbf x \) and \(\mathbf y \) belonging to the same triangle, \(u_h(\mathbf x )u_h(\mathbf y )\le K_1\mathbf x \mathbf y \) for some \(K_1>0\) independent of the mesh size h; (b) for any two points \(\mathbf x \) and \(\mathbf y \) in the same triangle, \(u_h(\mathbf x )u_h(\mathbf y )\le K_2\mathbf x \mathbf y \) for some \(K_2>0\) independent of the mesh size h; and (c) for any two points \(\mathbf x \) and \(\mathbf y \) on the mesh, \(u_h(\mathbf x )u_h(\mathbf y )\le K\mathbf x \mathbf y \) for some \(K>0\) independent of the mesh size h.
Due to the uniqueness of the viscosity solution [10, 11], the above proof shows that every subsequence of \(\{ u_h\}\) has a subsequence which converges uniformly to u. Therefore, \(\{ u_h\}\) converges uniformly to u as \(h\rightarrow 0\). This completes the proof.
In conclusion, we show that the FSM is convergent and its numerical solution converges to the viscosity solution as the grid size approaches zero.
3.3 Hopf formula and FSM
Here we show that, with piecewise linear approximation, the local solver based on the Hopf formula introduced in [7, 37] and the local solver, (2.7) + (2.12) + (2.11), of the FSM are equivalent. The subscript h is omitted in this part for notational simplicity.
Lemma 3.8
The local solver (3.8) based on the Hopf formula using piecewise linear approximation is equivalent to the local solver, (2.7) \(+\) (2.11) \(+\) (2.12), of the FSM.
Proof
Since \(G(\lambda )\) is convex in (0, 1), the global minimum of \(G(\lambda )\) on [0, 1] is obtained either at an interior point or at the two end points. If \(\lambda _M\) is one of the two end points, \(\{\lambda _M, \mathbf p (\lambda _M)\}\) is a global minimum point according to (2.7), (2.11) and (2.12). If \(\lambda _M \in (0,~1)\), since \(\frac{\mathrm{{d}}G(\lambda _M)}{\mathrm{{d}}\lambda }=0\), \(\{\lambda _M, \mathbf p (\lambda _M)\}\) is a global minimum point.
In conclusion, the local solver (3.8) based on the Hopf formula and that of the FSM, (2.7), (2.11) and (2.12), are equivalent to piecewise linear approximation. The proof is complete.
3.4 Local truncation error and error estimate
Lemma 3.1 shows the monotonicity and consistency of the local solver of the FSM, especially the contraction property (3.1). Here we study the relation for the errors at vertices of the same triangle. In contrast to error estimate for linear problems, the main difficulty for error estimate for nonlinear problems is that the errors do not satisfy the same equation as the solution.
3.5 Convergence study through contraction property
 Case 1:
The information propagates directly from the inflow boundary. Then the convergence can be obtained in a finite number of iterations independent of the grid size for Gauss–Seidel iterations that use upwind schemes with proper orderings, e.g., the fast sweeping method. For this example, one sweep is enough if the boundary condition is given on the left and bottom sides.
 Case 2:There is circular dependence, e.g., periodic boundary conditions.

Partial circular dependence, e.g., periodic boundary condition in x and given boundary condition on the bottom. This is equivalent to solving the PDE on a cylinder with boundary conditions given on the bottom. The characteristics wind from the bottom to the top of the cylinder. For this example, grid points on each horizontal line are coupled. Update at each grid point on the first line above the bottom side has the following error contraction from (3.21),In each sweeping with right orderings, the error at the first line is contracted by a factor of \(\alpha ^{\frac{1}{h}}\) after one iteration with \(\alpha =\frac{a}{a+b}<1\). Very few iterations, say n, are needed to get \({\left( \alpha ^{\frac{1}{h}}\right) }^n \backsim h^p\) for any p. Moreover, the smaller the h, the faster the convergence. The convergence due to the contraction follows similarly for other horizontal grid lines.$$\begin{aligned} e^n_{1,i}=\frac{a}{a+b}e^n_{1,i1}. \end{aligned}$$

Full circular dependence, e.g., periodic boundary conditions in both x and y with the value fixed at one point. This is equivalent to solving the PDE on a torus. The characteristics can be infinitely long or closed depending on a and b. If \(\frac{b}{a}\) is rational, each characteristic forms a closed curve. These closed curves are parallel to each other and do not have a starting point from which the information originates, except the characteristic that passes through the point with fixed value to make the solution unique. For the discrete system, all grid points are coupled together due to the circular dependence. In this case, the characteristic that passes through the given point will converge first and quickly. The solution has to compromise among the characteristics as well as propagate along the characteristics gradually due to the relaxation from the monotone scheme (3.20), i.e., the value at a grid point is the average of its neighbors’ values. When the grid is refined by half, there are twice as many characteristics as before involved. Hence the number of iterations is almost doubled as shown by numerical tests in Sect. 4. If \(\frac{b}{a}\) is irrational, the characteristic is infinitely long and covers the whole torus. Again all grid points are coupled together and the convergence is due to the relaxation from the scheme. When the grid is refined by half, the characteristic is twice as long as before with respect to the grid size. So the number of iterations is also doubled.

Remark 3.9
 1.
The ordering, the upwind finite difference scheme and the contraction rate are unknown á priori (all depending on the solution). That is why sweeping in four directions alternately is needed.
 2.
Nonlinear stability: for each point, due to the update with causality enforcement, e.g., accepting the smallest/best value propagated to this point so far, incorrect value at this point, no matter how large the error is, can be corrected when the correct information arrives during a sweeping with the right orderings; hence decay of the error may be better than geometric during iterations.
 3.
For the full periodic case, the characteristics behavior is much more complicated. The convergence depends on ergodicity of the characteristics on the torus. So when the mesh is refined, more iterations may or may not be needed for convergence as indicated by the numerical examples in Sect. 4.
Numerical examples are shown in the next section to verify the above study.
4 Numerical examples
In this section, we first use a few examples to show the general error estimate and the fast convergence. Then we use various tests on the linear convection equation as well as nonlinear Hamilton–Jacobi equations to demonstrate the convergence in different scenarios as discussed in Sect. 3.5. We record the number of iterations. One iteration means one sweep over all grid points.
 1.
We consider \(s \equiv 1\), and the domain is \([0, ~1]^2\). We first compute the distance to a source point. The boundary condition is imposed at the source point. The solution is not differentiable at the source, i.e., there has a source singularity. Table 1 shows the results. The accuracy is \(O(h\log h)\) due to the source singularity, which verifies the error estimate in [43]. Then we compute the distance to two disjoint circles of radius 0.1 and 0.15, respectively. Table 2 shows the results. We record maximum errors both inside and outside the circles. Although there are shocks both inside and outside the circles, they are different. Inside the circles, the shocks are located at the centers of the circles. All characteristics are converging to the centers. The second derivatives blow up like \(\frac{1}{d}\), where d is the distance to the centers. So the accumulation of local truncation error behaves in the same way as near a source singularity, which gives an error estimate of the form \(O(h\log h)\). Outside the circles, the shock is a smooth curve (at the equal distance locations). As long as the shock is a smooth curve without end points, the second derivatives are uniformly bounded except at the shock. So the local truncation error is still \(O(h^2)\) all the way up to the shock. At the shock, the local truncation error is O(h). So the global error is still O(h), which is verified by the numerical tests.
 2.
We consider \(s(x,y) = 0.5  (y0.25)\) and the domain is \([0,~0.5]^2\). For pointsource conditions, the exact solution can be derived as in [17]. Table 3 shows the results with one point source. When the boundary condition is enforced only at the source, the maximum error is \(O(h\log h)\) since the solution has a source singularity. When the exact solution is enforced on a disk of radius 0.1 centered at the source and the computation is performed outside the disk, then the maximum error is O(h) since the solution is smooth outside the disk. Table 4 shows the results with two source points and with the same treatment at the sources. Similar accuracy is observed.
Example 1: Case 1 with distance to one source point
Source point \(=(0.5,0.5)\), domain = \([0,~1]^2\)  
Mesh  \(21 \times 21\)  \(41\times 41\)  \(81\times 81\)  \(161\times 161\)  \(321\times 321\) 
\(uu_h_\infty \)  4.11E−2  2.56E−2  1.55E−2  9.19E−3  5.33E−3 
Convergence order  –  0.6804  0.7221  0.7572  0.7865 
\(\frac{uu_h_\infty }{h\log h}\)  0.2741  0.2778  0.2835  0.2897  0.2955 
#Iter  5  5  5  5  5 
Example 1: Case 1 with distance to two circles
Circle 1 with center (0.2, 0.2) and radius 0.1  
Circle 2 with center (0.8, 0.8) and radius 0.15  
Domain \(= [0,~1]^2\)  
Mesh  \(21 \times 21\)  \(41\times 41\)  \(81\times 81\)  \(161\times 161\)  \(321\times 321\) 
\(uu_h_\infty \) outside circles  1.76E−2  1.09E−2  5.66E−3  2.95E−3  1.50E−3 
Convergence order  –  0.6997  0.9392  0.9432  0.9747 
\(uu_h_\infty \) inside circles  2.03E−2  1.33E−2  8.33E−3  5.08E−3  3.03E−3 
Convergence order  –  0.6059  0.6807  0.7141  0.7442 
\(\frac{uu_h_\infty }{h\log h}\) inside circles  0.1356  0.1447  0.1520  0.1600  0.1681 
#Iter  8  8  8  8  8 
Example 1: Case 2 with one source point
Boundary condition assigned at the source  
Mesh  \(51\times 51\)  \(101\times 101\)  \(201\times 201\)  \(4011\times 401\)  \(801\times 801\) 
\(uu_h_\infty \)  3.06E−2  1.75E−2  9.87E−3  5.52E−3  3.06E−3 
Convergence order  –  0.8062  0.8262  0.8384  0.8511 
\(\frac{uu_h_\infty }{h\log h}\)  0.3911  0.3800  0.3725  0.3685  0.3662 
#Iter  8  8  8  8  8 
Boundary condition assigned at the disk of radius 0.1 centered at the source  
Mesh  \(51\times 51\)  \(101\times 101\)  \(201\times 201\)  \(4011\times 401\)  \(801\times 801\) 
\(uu_h_\infty \)  1.73E−2  8.66E−3  4.32E−3  2.16E−3  1.08E−3 
Convergence order  –  0.9983  1.0033  1.0000  1.0000 
#Iter  8  8  8  8  8 
Example 1: Case 2 with two source points
Boundary condition assigned at the sources  
Mesh  \(51\times 51\)  \(101\times 101\)  \(201\times 201\)  \(4011\times 401\)  \(801\times 801\) 
\(uu_h_\infty \)  2.95E−2  1.74E−2  1.02E−2  5.83E−3  3.30E−3 
Convergence order  –  0.7616  0.7705  0.8070  0.8210 
\(\frac{uu_h_\infty }{h\log h}\)  0.3770  0.3778  0.3850  0.3892  0.3949 
#Iter  8  8  8  8  8 
Boundary condition assigned at the disk of radius 0.1 centered at the sources  
Mesh  \(51\times 51\)  \(101\times 101\)  \(201\times 201\)  \(4011\times 401\)  \(801\times 801\) 
\(uu_h_\infty \)  1.55E−2  7.72E−3  3.84E−3  1.92E−3  9.58E−4 
Convergence order  –  1.0056  1.0075  1.0000  1.0030 
#Iter  8  8  8  8  8 
Example 2: Linear problem with different boundary conditions (B.C.)
B.C. \(u(x,0)=\sin (2\pi x),~u(0,y)=\sin (2\pi \frac{a}{b}y)\), domain \(=[0,~1]^2\)  
# iter  \(a =\) 1, \(b =\) 2  \(a =\) 2, \(b =\) 1  \(a =\) 4, \(b =\) 1  \(a =\) 8, \(b =\) 1 
\(h =\) 1/20  2  2  2  2 
\(h =\) 1/40  2  2  2  2 
\(h =\) 1/80  2  2  2  2 
\(h =\) 1/160  2  2  2  2 
B.C. \(u(x,0)=\sin (2\pi x)\), periodic in x, domain \(=[0,~1]^2\)  
# iter  \(a =\) 1, \(b =\) 2  \(a =\) 2, \(b =\) 1  \(a =\) 4, \(b =\) 1  \(a =\) 8, \(b =\) 1 
\(h =\) 1/20  6  13  23  46 
\(h =\) 1/40  5  9  16  32 
\(h =\) 1/80  4  7  12  23 
\(h =\) 1/160  4  6  10  18 
B.C. periodic in both x and y, \(u(0,0)=0\), \(a=1, b=2\), domain \(=[0,~1]^2\)  
Mesh  \(h=1/20\)  \(h=1/40\)  \(h=1/80\)  \(h=1/160\) 
# iter  923  1917  3908  7880 
B.C. periodic in both x and y, \(u(0,0)=0\), \(a=e, b=\pi \), domain \(=[0,~1]^2\)  
Mesh  \(h=1/20\)  \(h=1/40\)  \(h=1/80\)  \(h=1/160\) 
# iter  968  1949  3829  7527 
 (a)without spatial variation.$$\begin{aligned}M_0=\left( \begin{array}{cc}1/4 &{} \quad 0\\ 0 &{} \quad 1\\ \end{array}\right) , \end{aligned}$$
 (b)with spatial variation.$$\begin{aligned}M_0=\left( \begin{array}{cc}1/4(10.5\sin \pi x \sin \pi y) &{} \quad 0\\ 0 &{} \quad 1+0.5\sin \pi x \sin \pi y\\ \end{array}\right) , \end{aligned}$$
Example 3: Case 1
# iter  \(\varvec{\theta }{} \mathbf = \varvec{0}\)  \(\varvec{\theta }{} \mathbf = {} \mathbf 5 \varvec{\pi }/\mathbf 6 \)  \(\varvec{\theta }{} \mathbf = {} \mathbf 2 \varvec{\pi }/\mathbf 3 \)  \(\varvec{\theta }{} \mathbf = \varvec{\pi }/\mathbf 2 \)  \(\varvec{\theta }{} \mathbf = \varvec{\pi }/\mathbf{3}\)  \(\varvec{\theta }\mathbf{=}\varvec{\pi }/\varvec{6}\) 

\(M_0\) without spatial variation, \(\alpha =0.15\), tolerance \(=10^{12}\)  
\(h=\) 1/320  2  2  2  2  2  2 
\(h=\) 1/640  2  2  2  2  2  2 
\(h=\) 1/1280  2  2  2  2  2  2 
\(M_0\) with spatial variation, \(\alpha =0.15\), tolerance \(=10^{12}\)  
\(h=\) 1/320  2  2  2  2  2  2 
\(h=\) 1/640  2  2  2  2  2  2 
\(h=\) 1/1280  2  2  2  2  2  2 
 1.With Dirichlet boundary conditionFor this case, the characteristics from the boundary are oriented in one direction, either from lower left to upper right or from lower right to upper left, which can be predetermined. Hence one ordering is needed for convergence. Table 6 shows the ordering and the test results.$$\begin{aligned} u(x,0)=(\alpha +1)  \alpha \cos (\pi x). \end{aligned}$$
 2.With Dirichlet boundary conditionand partial periodic boundary condition: periodic in x. For this case, the characteristics from the boundary are oriented in two directions: from lower left to upper right and from lower right to upper left. Hence two orderings are needed for convergence. Table 7 shows the orderings and the test results. The number of iterations does not increase as the mesh is refined.$$\begin{aligned} u(x,0)=(\alpha +1)  \alpha \cos (2\pi x), \end{aligned}$$
 3.With pointsource conditionFor this case, the characteristics from the source point are oriented in all directions. Hence four orderings (\(i=1{:}I,~j=1{:}J;~i=1{:}I,~j=J{:}1; ~i=I{:}1,~j=1{:}J; i=I{:}1,~j=J{:}1\)) are needed for convergence. Table 8 shows the orderings and the test results. The number of iterations does not increase as the mesh is refined.$$\begin{aligned} u(1/2,1/2)= 0. \end{aligned}$$
 4.With partial periodic boundary condition: u is periodic in x andFor this case, the characteristics from the source point are oriented in all directions. Hence four orderings (\(i=1{:}I,j=1{:}J;~i=1{:}I,j=J{:}1; ~i=I{:}1,j=1{:}J; ~i=I{:}1,j=J{:}1\)) are needed for convergence. Table 9 shows the orderings and the test results. The number of iterations does not increase as the mesh is refined.$$\begin{aligned} u(1/2,1/2)= 0. \end{aligned}$$
 5.With full partial periodic boundary condition: u is periodic in x and y, andFor this case, the characteristics from the source point are oriented in all directions. Hence four orderings (\(i=1{:}I,j=1{:}J;~i=1{:}I,j=J{:}1; ~i=I{:}1,j=1{:}J;~ i=I{:}1,j=J{:}1\)) are needed for convergence. Table 10 shows the orderings and the test results. The number of iterations does not increase as the mesh is refined, which is different from the linear case of Example 3 in that the characteristics originated at the source propagate in all directions.$$\begin{aligned} u(1/2,1/2)= 0. \end{aligned}$$
Example 3: Case 2
# iter  \(\varvec{\theta }\varvec{=}\varvec{0}\)  \(\varvec{\theta }\varvec{=}\varvec{5}\varvec{\pi }/\varvec{6}\)  \(\varvec{\theta }\varvec{=}\varvec{2}\varvec{\pi }/\varvec{3}\)  \(\varvec{\theta }\varvec{=}\varvec{\pi }/\varvec{2}\)  \(\varvec{\theta }\varvec{=}\varvec{\pi }/\varvec{3}\)  \(\varvec{\theta }\varvec{=}\varvec{\pi }/\varvec{6}\) 

\(M_0\) without spatial variation, \(\alpha =0.15\), tolerance \(=10^{12}\)  
\(h=\) 1/320  3  6  6  3  7  7 
\(h=\) 1/640  3  6  6  3  7  7 
\(h=\) 1/1280  3  6  6  3  7  7 
\(M_0\) with spatial variation, \(\alpha =0.15\), tolerance \(=10^{12}\)  
\(h=\) 1/320  5  6  8  3  7  7 
\(h=\) 1/640  5  6  8  3  7  7 
\(h=\) 1/1280  5  6  8  3  7  7 
Example 3: Case 3
# iter  \(\varvec{\theta }\varvec{=}\varvec{0}\)  \(\varvec{\theta }\varvec{=}\varvec{5}\varvec{\pi }/\varvec{6}\)  \(\varvec{\theta }\varvec{=}\varvec{2}\varvec{\pi }/\varvec{3}\)  \(\varvec{\theta }\varvec{=}\varvec{\pi }/\varvec{2}\)  \(\varvec{\theta }\varvec{=}\varvec{\pi }/\varvec{3}\)  \(\varvec{\theta }\varvec{=}\varvec{\pi }/\varvec{6}\) 

\(M_0\) without spatial variation, \(\alpha =0.15\), tolerance \(=10^{12}\)  
\(h=\) 1/320  5  5  5  5  5  5 
\(h=\) 1/640  5  5  5  5  5  5 
\(h=\) 1/1280  5  5  5  5  5  5 
\(M_0\) with spatial variation, \(\alpha =0.15\), tolerance \(=10^{12}\)  
\(h=\) 1/320  7  8  8  8  6  6 
\(h=\) 1/640  7  8  8  8  6  6 
\(h=\) 1/1280  7  8  8  8  6  6 
Example 3: Case 4
# iter  \(\varvec{\theta }\varvec{=}\varvec{0}\)  \(\varvec{\theta }\varvec{=}\varvec{5}\varvec{\pi }/\varvec{6}\)  \(\varvec{\theta }\varvec{=}\varvec{2}\varvec{\pi }/\varvec{3}\)  \(\varvec{\theta }\varvec{=}\varvec{\pi }/\varvec{2}\)  \(\varvec{\theta }\varvec{=}\varvec{\pi }/\varvec{3}\)  \(\varvec{\theta }\varvec{=}\varvec{\pi }/\varvec{6}\) 

\(M_0\) without spatial variation, \(\alpha =0.15\), tolerance \(=10^{12}\)  
\(h=\) 1/320  5  24  23  6  21  23 
\(h=\) 1/640  5  24  23  6  21  23 
\(h=\) 1/1280  5  24  23  6  21  22 
\(M_0\) with spatial variation, \(\alpha =0.15\), tolerance \(=10^{12}\)  
\(h=\) 1/320  8  28  21  8  20  26 
\(h=\) 1/640  8  28  21  8  20  26 
\(h=\) 1/1280  8  28  21  8  20  26 
Example 3: Case 5
# iter  \(\varvec{\theta }\varvec{=}\varvec{0}\)  \(\varvec{\theta }\varvec{=}\varvec{5}\varvec{\pi }/\varvec{6}\)  \(\varvec{\theta }\varvec{=}\varvec{2}\varvec{\pi }/\varvec{3}\)  \(\varvec{\theta }\varvec{=}\varvec{\pi }/\varvec{2}\)  \(\varvec{\theta }\varvec{=}\varvec{\pi }/\varvec{3}\)  \(\varvec{\theta }\varvec{=}\varvec{\pi }/\varvec{6}\) 

\(M_0\) without spatial variation, \(\alpha =0.15\), tolerance \(=10^{12}\)  
\(h=\) 1/320  5  25  31  5  30  23 
\(h=\) 1/640  7  25  31  6  30  24 
\(h=\) 1/1280  7  24  30  7  29  23 
\(M_0\) with spatial variation, \(\alpha =0.15\), tolerance \(=10^{12}\)  
\(h=\) 1/320  8  27  21  8  20  25 
\(h=\) 1/640  8  27  21  9  20  25 
\(h=\) 1/1280  9  27  21  10  20  24 
5 Conclusion
We investigate the convergence of iterative methods for hyperbolic problems such as the boundary value problems of static convex Hamilton–Jacobi equations using the fast sweeping method as an example. We prove the convergence and show that the contraction property of monotone upwind schemes combined with Gauss–Seidel iterations and proper orderings can provide fast convergence for such hyperbolic problems. The mechanism is different from that for elliptic problems and may render different behavior when the mesh is refined. The study is verified by various numerical examples.
Acknowledgements
S. Luo is partially supported by NSF Grant DMS 1418908. H. Zhao is partially supported by NSF Grant DMS 1418422.
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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