Abstract
Our analysis has so far been confined to scalar conservation laws in one dimension. Clearly, the multidimensional case is considerably more important. Luckily enough, the analysis in one dimension can be carried over to higher dimensions by essentially treating each dimension separately. This technique is called dimensional splitting. The final results are very much the natural generalizations one would expect.
The same splitting techniques of dividing complicated differential equations into several simpler parts can in fact be used to handle other problems. These methods are generally called operator splitting methods or fractional steps methods.
Just send me the theorems, then I shall find the proofs.Lucky guy! Paraphrased from Diogenes Laertius, Lives of Eminent Philosophers, c. a.d. 200. — Chrysippus told Cleanthes, 3rd century BC Just send me the theorems, then I shall find the proofs. — Chrysippus told Cleanthes, 3rd century BC Lucky guy! Paraphrased from Diogenes Laertius, Lives of Eminent Philosophers, c. a.d. 200.
Just send me the theorems, then I shall find the proofs.
— Chrysippus told Cleanthes, 3rd century BC
Lucky guy! Paraphrased from Diogenes Laertius, Lives of Eminent Philosophers, c. a.d. 200.
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Our analysis has so far been confined to scalar conservation laws in one dimension. Clearly, the multidimensional case is considerably more important. Luckily enough, the analysis in one dimension can be carried over to higher dimensions by essentially treating each dimension separately. This technique is called dimensional splitting. The final results are very much the natural generalizations one would expect.
The same splitting techniques of dividing complicated differential equations into several simpler parts can in fact be used to handle other problems. These methods are generally called operator splitting methods or fractional steps methods.
4.1 Dimensional Splitting Methods
We will show in this section how one can analyze scalar multidimensional conservation laws by dimensional splitting, which amounts to solving one space direction at a time. To be more concrete, let us consider the two-dimensional conservation law
If we let \(S^{f,x}_{t}u_{0}\) denote the solution of
(where y is a passive parameter), and similarly let \(S^{g,y}_{t}u_{0}\) denote the solution of
(x is a parameter), then the idea of dimensional splitting is to approximate the solution of (4.1) as follows:
⋄ Example 4.1 (A single discontinuity)
We first show how this works on a concrete example. Let
and
with \(u_{r}> u_{l}\). The solution in the x-direction for fixed y gives a rarefaction wave, the left and right parts moving with speeds u l and u r , respectively. With a quadratic flux, the rarefaction wave is a linear interpolation between the left and right states. Thus
The solution in the y-direction for fixed x with initial state \(u^{1/2}\) will exhibit a focusing of characteristics. The left state, which now equals u r , will move with speed given by the derivative of the flux function, in this case u r , and hence overtake the right state, given by u l , which moves with smaller speed, namely u l . The characteristics interact at a time t given by
or \(t={\Updelta t}\). At that time we are back to the original Riemann problem between states u l and u r at the point \(x=y\). Thus
Another application of \(S^{f,x}_{{\Updelta t}}\) will of course give
So we have that \(u^{n}=u_{0}\) for all \(n\in\mathbb{N}\). When we introduce coordinates
the equation transforms into
We see that \(u(x,y,t)=u_{0}(x,y)\), and consequently \(\lim_{\Updelta t\to 0}u^{n}=u_{0}\) (where we keep \(n{\Updelta t}=t\) fixed). Thus the dimensional splitting procedure produces approximate solutions converging to the right solution in this case. ⋄
We will state all results for the general case of arbitrary dimension, while proofs will be carried out in two dimensions only, to keep the notation simple. We first need to define precisely what is meant by a weak entropy solution of the initial value problem
where \(f = (f_{1},\dots,f_{m})\), and the spatial variables are denoted by \((x_{1},\dots,x_{m}) \in \mathbb{R}^{m} \). Here we adopt the Kružkov entropy condition from Chapt. 2, and say that u is a (weak) Kružkov entropy solution of (4.3) for time \([0,T]\) if u is a bounded function that satisfiesFootnote 1
for all constants \(k\in\mathbb{R}\) and all nonnegative test functions \(\varphi\in C_{0}^{\infty}(\mathbb{R}^{m}\times[0,T])\). It certainly follows as in the one-dimensional case that u is a weak solution, i.e.,
for all test functions \(\varphi\in C_{0}^{\infty}(\mathbb{R}^{m}\times[0,\infty))\).
Our analysis aims at two different goals. We first show that the dimensional splitting indeed produces a sequence of functions that converges to a solution of the multidimensional equation (4.3). Our discussion will here be based on the more or less standard argument using Kolmogorov’s compactness theorem. The argument is fairly short. In order to obtain stability in the multidimensional case in the sense of Theorem 2.14, we show that dimensional splitting preserves this stability. Furthermore, we show how one can use front tracking as our solution operator in one dimension in combination with dimensional splitting. Finally, we determine the appropriate convergence rate of this procedure. This analysis strongly uses Kuznetsov’s theory from Sect. 3.3, but matters are more complicated and technical than in one dimension.
We shall now show that dimensional splitting produces a sequence that converges to the entropy solution u of (4.3); that is, the limit u should satisfy (4.4). As promised, our analysis will be carried out in the two-dimensional case only, i.e., for equation (4.1). Assume that u 0 is a function in \(L^{1}(\mathbb{R}^{2})\cap L^{\infty}(\mathbb{R}^{2})\cap{BV}(\mathbb{R}^{2})\) (consult Definition A.2 for a definition of \({BV}(\mathbb{R}^{2})\); see also (A.11)). Let \(t_{n}=n{\Updelta t}\) and \(t_{n+1/2}=\big(n+\frac{1}{2}\big){\Updelta t}\). Define
for \(n\in\mathbb{N}_{0}\). We shall also be needing an approximate solution for \(t\neq t_{n}\). We want the approximation to be an exact solution to a one-dimensional conservation law in each interval \(\left[t_{j},t_{j+1/2}\right]\), \(j=k/2\), and \(k\in\mathbb{N}_{0}\). The way to do this is to make ‘‘time go twice as fast’’ in each such interval; i.e., let \(u_{{\Updelta t}}\) be defined byFootnote 2
We will use Theorem A.11, that is, we show that the sequence \(\{u_{{\Updelta t}}\}\) is compact. Since neither the operator \(S^{f,x}\) nor \(S^{g,y}\) increases the \(L^{\infty}\) norm, \(u_{{\Updelta t}}\) will be uniformly bounded, i.e.,
independent of \({\Updelta t}\).
Next we study the total variation. We start by considering
where we used Lemma A.1 and the L 1-contractivity; cf. Theorem 2.15 (vi). The interchange of integrals and limits is justified using Lebesgue’s dominated convergence theorem.
For the solution constructed from dimensional splitting we have
using the TVD property of \(S^{f,x}\) and (4.9). Similarly,
and thus
follows by induction. This extends to
We now want to establish Lipschitz continuity in time of the L 1-norm, i.e.,
for some constant C. By repeated use of the triangle inequality it suffices to estimate
Using Theorem 2.15 (vi), we conclude that the first term in (4.13) is bounded by \(\|f\|_{\mathrm{Lip}}{\Updelta t}\,\mathrm{T.V.}_{{x,y}}\left(u^{n}\right)\). For the second term. we obtain, using in addition (4.9), the bound \(\|g\|_{\mathrm{Lip}}{\Updelta t}\,\mathrm{T.V.}_{{x,y}}\left(u^{n}\right)\). This proves
Using interpolation, we obtain the estimate
where \(t\in[t_{n},t_{n+1})\) and \(s\in[t_{m},t_{m+1})\).
Using Theorem A.11, we conclude the existence of a convergent subsequence, also labeled \(\left\{u_{{\Updelta t}}\right\}\), and set \(u=\lim_{{\Updelta t}\to 0}u_{{\Updelta t}}\). Next we have to prove that the limit u is a weak entropy solution.
Let \(\phi=\phi(x,y,t)\) be a nonnegative test function, and define \(\varphi\) by \(\varphi(x,y,t)=\phi(x,y,\frac{1}{2}t+t_{n})\). By defining \(\tau=2(t-n{\Updelta t})\), we have that for each y, the function \(u_{{\Updelta t}}\) is a weak solution in x on the strip \(t\in[t_{n},t_{n+1/2}]\) satisfying the inequality
for all constants k. Here \(q^{f}(u,k)=\mathrm{sign}\left(u-k\right)(f(u)-f(k))\). Changing back to the t variable, we find that
Similarly,
Here q g is defined similarly to q f, using g instead of f. Integrating (4.17) over y and (4.18) over x and adding the two results and summing over n, we obtain
where \(\chi_{n}\) and \(\tilde{\chi}_{n}\) denote the characteristic functions of the strips \(t_{n}\leq t\leq t_{n+1/2}\) and \(t_{n+1/2}\leq t\leq t_{n+1}\), respectively. As \({\Updelta t}\) tends to zero, it follows that
Specifically, for continuous functions ψ we see that
(where \(t_{n}^{*}\) is in \([t_{n},t_{n+1/2}]\)), by definition of the Riemann integral. The general case follows by approximation.
Letting \({\Updelta t}\to 0\), we thus obtain
which proves that \(u(x,y,t)\) is a solution to (4.1) satisfying the Kružkov entropy condition.
Next, we want to prove uniqueness of solutions of multidimensional conservation laws. Let u and v be two Kružkov entropy solutions of the conservation law
with initial data u 0 and v 0, respectively. The argument in Sect. 2.4 leads, with no fundamental changes in the multidimensional case, to the same result (2.65), namely,
thereby proving uniqueness. Using the fact that if every subsequence of a sequence has a further subsequence converging to the same limit, the whole sequence converges to that (unique) limit, we find that the whole sequence \(\left\{u_{{\Updelta t}}\right\}\) converges, not just a subsequence. We have proved the following result.
Theorem 4.2
Let f j be piecewise twice continuously differentiable functions, and furthermore, let u 0 be an integrable and bounded function in \({BV}(\mathbb{R}^{m})\). Define the sequence of functions \(\{u^{n}\}\) by \(u^{0}=u_{0}\) and
Introduce the function (where \(t_{r}=r{\Updelta t}\) for a rational number r)
for \(t\in[t_{n+(j-1)/m},t_{n+j/m}]\). Fix T > 0. Then for every sequence \(\left\{{\Updelta t}\right\}\) such that \({\Updelta t}\to 0\), for all \(t\in[0,T]\) the function \(u_{{\Updelta t}}(t)\) converges to the unique weak solution \(u(t)\) of (4.3) satisfying the Kružkov entropy condition (4.4). The limit is in \(C([0,T];L^{1}_{\mathrm{loc}}(\mathbb{R}^{m}))\).
To prove stability of the solution with respect to flux functions, we will show that the one-dimensional stability result (2.80) in Sect. 2.4 remains valid with obvious modifications in several dimensions. Let u and v denote the unique solutions of
and
respectively, that satisfy the Kružkov entropy condition. We want to estimate the L 1-norm of the difference between the two solutions. To this end, we first consider
Next we employ the trivial, but useful, inequality
Thus
which implies
Consider next \(t\in[t_{n},t_{n+1/2})\). Then the continuous interpolants defined by (4.7) satisfy
Observe that the above argument also holds mutatis mutandis in the general case of a scalar conservation law in any dimension. We summarize our results in the following theorem.
Theorem 4.3
Let u 0 be in \(L^{1}(\mathbb{R}^{m})\cap L^{\infty}(\mathbb{R}^{m})\cap{BV}(\mathbb{R}^{m})\), and let f j be piecewise twice continuously differentiable functions for \(j=1,\dots,m\), and set \(f=(f_{1},\dots,f_{m})\). Then there exists a unique solution \(u=u(x_{1},\dots,x_{m},t)\) of the initial value problem
that satisfies the Kružkov entropy condition (4.4). The solution satisfies
Furthermore, if v 0 and g share the same properties as u 0 and f, respectively, then the unique weak Kružkov entropy solution of
satisfies
If \(u_{0}\leq v_{0}\) and \(f=g\), then also \(u\leq v\) on all of \(\mathbb{R}^{m}\times[0,\infty)\).
Proof
The proof of the Lipschitz continuity in time follows from (4.15). The monotonicity statement at the end follows using the L 1-contractivity (the special case of (4.26) with \(f=g\)) as in the one-dimensional case by employing the Crandall–Tartar lemma. □
(See also Exercise 4.1.)
4.2 Dimensional Splitting and Front Tracking
It doesn’t matter if the cat is black or white. As long as it catches rats, it’s a good cat. — Deng Xiaoping (1904–1997)
In this section we will study the case in which we use front tracking to solve the one-dimensional conservation laws. More precisely, we replace the flux functions f and g (in the two-dimensional case) by piecewise linear continuous interpolations \(f_{\delta}\) and \(g_{\delta}\), with the interpolation points spaced a distance δ apart. The aim is to determine the convergence rate toward the solution of the full two-dimensional conservation law as \(\delta\to 0\) and \({\Updelta t}\to 0\).
With the front-tracking approximation, the one-dimensional solutions will be piecewise constant if the initial condition is piecewise constant. In order to prevent the number of discontinuities from growing without bound, we will project the one-dimensional solution \(S^{f_{\delta},x}u\) onto a fixed grid in the \((x,y)\)-plane before applying the operator \(S^{g_{\delta},y}\).
To be more concrete, let the grid spacing in the x- and y- directions be given by \(\Updelta x\) and \(\Updelta y\), respectively, and let I ij denote the grid cell
The projection operator π is defined by
Let the approximate solution at the discrete times t l be defined as
for \(n=0,1,2,\dots\), with \(u^{0}=\pi u_{0}\). We collect the discretization parameters in \(\eta=(\delta,\Updelta x,\Updelta y,{\Updelta t})\). In analogy to (4.7), we define \(u_{\eta}\) as
In Fig. 4.1 we illustrate how this works. Starting in the upper left corner, the operator \(S^{f_{\delta},x}_{{\Updelta t}}\) takes us to the upper right corner; then we apply π and move to the lower right corner. Next, \(S^{g_{\delta},y}_{{\Updelta t}}\) takes us to the lower left corner, and finally π takes us back to the upper left corner, this time with n incremented by 1.
Front tracking and dimensional splitting on a \(3\times 3\) grid
To prove that \(u_{\eta}\) converges to the unique solution u as \(\eta\to 0\), we essentially mimic the approach we just used to prove Theorem 4.2. First of all we observe that
since \(S^{f_{\delta},x}\), \(S^{g_{\delta},y}\), and π all obey a maximum principle. On each rectangle I ij the function \(u_{\eta}\) is constant for \(t={\Updelta t}\). In a desperate attempt to simplify the notation, we write
Next we go carefully through one full time step in this construction, starting with \(u_{ij}^{n}\). At each step we define a shorthand notation that we will use in the estimates. When we consider \(u_{ij}^{n}\) as a function of x only, we write
(The argument ‘‘0’’ on the left-hand side indicates the start of the time variable before we advance time an interval \({\Updelta t}\) using \(S^{f_{\delta},x}_{{\Updelta t}}\).) Advancing the solution in time by \({\Updelta t}\) by applying front tracking in the x-variable produces
(The x-dependence is suppressed in the notation on the left-hand side.) We now apply the projection π, which yields
After this sweep in the x-variable, it is time to do the y-direction. Considering \(u_{ij}^{n+1/2}\) as a function of y, we write
to which we apply the front-tracking solution operator in the y-direction
(The y-dependence is suppressed in the notation on the left-hand side.) One full time step is completed by a final projection
Using this notation, we first want to prove that the total variation is bounded in the sense that
We will show that
an analogous argument gives \(\mathrm{T.V.}\left(u^{n+1}\right)\leq\mathrm{T.V.}\left(u^{n+1/2}\right)\), from which we conclude that
and (4.29) follows by induction. By definition,
while
We first consider
where we first used that \(\mathrm{T.V.}_{x}\left(\pi\phi\right)\leq\mathrm{T.V.}_{x}\left(\phi\right)\) for step functions ϕ. This follows from the following argument: Let \(\phi_{c}\) be a continuous function equal to ϕ except close to each jump, where we use a linear interpolation. Then \(\mathrm{T.V.}_{x}\left(\phi\right)=\mathrm{T.V.}_{x}\left(\phi_{c}\right)\geq\mathrm{T.V.}_{x}\left(\pi\phi\right)\), since \(\pi\phi\) is just a particular partition of \(\phi_{c}\); cf. (A.1). Subsequently we used that \(\mathrm{T.V.}\left(v\right)\leq\mathrm{T.V.}\left(v_{0}\right)\) for solutions v of one-dimensional conservation laws with initial data v 0. For the second term in the definition of \(\mathrm{T.V.}\left(u^{n+1/2}\right)\) we obtain (cf. (4.10))
The first inequality follows from \(\left|\pi\phi\right|\leq\pi\left|\phi\right|\); thereafter, we use \(\int_{I_{ij}}\pi\phi=\int_{I_{ij}}\phi\), and finally we use the L 1-contractivity, \({\left\|v-w\right\|}_{L^{1}(\mathbb{R})}\leq{\left\|v_{0}-w_{0}\right\|}_{L^{1}(\mathbb{R})}\), of solutions of one-dimensional conservation laws. Multiplying (4.33) by \(\Updelta y\), summing over j, dividing (4.34) by \(\Updelta x\), and finally adding the results gives (4.30).
Finally, we want to show the analogue of Lipschitz continuity in time of the spatial L 1-norm as expressed in (4.12). We want to prove the following result:
To prove (4.35), it suffices to show that
We start by writing
Integrating this inequality over \(\mathbb{R}^{2}\) gives
We see that two terms involve the projection operator π. For these terms we prove the estimate
We will prove (4.38) in the one-dimensional case only (See Exercise 4.3). Consider (where \(I_{i}=[x_{i},x_{i+1})\))
For the two remaining terms in (4.37) we obtain, using the Lipschitz continuity in time in the L 1 norm in the x-variable (see Theorem 2.15), that
Combining this result with (4.29), (4.38), we conclude that (4.36), and hence also (4.35), holds.
So far we have obtained the following estimates:
- (i):
-
Uniform boundedness,
$$\displaystyle{\left\|u_{\eta}(t)\right\|}_{L^{\infty}(\mathbb{R}^{2})}\leq{\left\|u^{0}\right\|}_{L^{\infty}(\mathbb{R}^{2})}.$$ - (ii):
-
Uniform bound on the total variation,
$$\displaystyle\mathrm{T.V.}\left(u^{n}\right)\leq\mathrm{T.V.}\left(u_{0}\right).$$ - (iii):
-
Lipschitz continuity in time,
$$\displaystyle\begin{aligned}\displaystyle{\left\|u_{\eta}(t_{m})-u_{\eta}(t_{n})\right\|}_{L^{1}(\mathbb{R}^{2})}&\displaystyle\leq\Bigl(\max\{\,\|f_{\delta}\|_{\mathrm{Lip}},\|g_{\delta}\|_{\mathrm{Lip}}\,\}+2\frac{{\Updelta x}+\Updelta y}{{\Updelta t}}\Bigr)\\ \displaystyle&\displaystyle\quad\times\mathrm{T.V.}\left(u^{0}\right)\left|t_{m}-t_{n}\right|.\end{aligned}$$(4.41)
From Theorem A.11 we conclude that the sequence \(\{u_{\eta}\}\) has a convergent subsequence as \(\eta\to 0\), provided that the ratio \(\max\{\Updelta x,\Updelta y\}/{\Updelta t}\) remains bounded. We let u denote its limit. Furthermore, this sequence converges in \(C([0,T];L^{1}_{\mathrm{loc}}(\mathbb{R}^{2}))\) for every positive T.
It remains to prove that the limit is indeed an entropy solution of the full two-dimensional conservation law. We first use that \(u_{j}^{n}(x,t)\) (suppressing the y-dependence) is a solution of the one-dimensional conservation law in the time interval \([t_{n},t_{n+1/2}]\). Hence we know that
Similarly, we obtain for the y-direction
Integrating the first inequality over y and the second over x and adding the results as well as adding over n gives, where \(T=N{\Updelta t}\),
and as before, \(\chi_{n}\) and \(\tilde{\chi}_{n}\) denote the characteristic functions on \(\{(x,y,t)\mid t\in[t_{n},t_{n+1/2}]\}\) and \(\{(x,y,t)\mid t\in[t_{n+1/2},t_{n+1}]\}\), respectively. Observe that we have obtained the right-hand side by using a projection at each time step. As \(n\to\infty\) and \({\Updelta t}\to 0\) while keeping T fixed, we have that \(\sum_{n}\chi_{n}\overset{*}{\rightharpoonup}\frac{1}{2}\). To estimate the right-hand side we first observe that
and since the absolute value function is convex, Jensen’s inequality implies that
Thus we obtain
using (4.42). This implies
since
where we have used (4.38). Thus
In order to conclude that u is an entropy solution, we need the right-hand side of (4.43) to vanish as \({\Updelta x},\Updelta y,{\Updelta t}\to 0\); that is, we need to assume that
as \(\eta\to 0\). Under this assumption,
which shows that u indeed satisfies the Kružkov entropy condition. We summarize the result.
Theorem 4.4
Let u 0 be an integrable and bounded function in \(L^{\infty}(\mathbb{R}^{m})\cap{BV}(\mathbb{R}^{m})\), and let f j be piecewise twice continuously differentiable functions for \(j=1,\dots,m\). Construct an approximate solution \(u_{\eta}\) using front tracking by defining
and
where \(x=\left(x_{1},\dots,x_{m}\right)\).
For every sequence \(\left\{\eta\right\}\), with \(\eta=\left(\Updelta x_{1},\dots,\Updelta x_{m},\Updelta t,\delta\right)\), where \(\eta\to 0\) and
we have that \(\{u_{\eta}\}\) converges to the unique solution \(u=u(x,t)\) of the initial value problem
which satisfies the Kružkov entropy condition.
4.3 Convergence Rates
Now I think I’m wrong on account of those damn partial integrations. I oscillate between right and wrong. — Letter from Feynman to Welton (1936)
In this section we show how fast front tracking plus dimensional splitting converges to the exact solution. The analysis is based on Kuznetsov’s lemma.
We start by generalizing Kuznetsov’s lemma, Theorem 3.14, to the present multidimensional setting. Although the argument carries over, we will present the relevant definitions in arbitrary dimension.
Let the class \(\mathcal{K}\) consist of maps \(u\colon[0,\infty)\to L^{1}(\mathbb{R}^{m})\cap{BV}(\mathbb{R}^{m})\cap L^{\infty}(\mathbb{R}^{m})\) such that:
-
(i)
The limits \(u(t\pm)\) exist.
-
(ii)
The function u is right continuous, i.e., \(u(t+)=u(t)\).
-
(iii)
\({\left\|u(t)\right\|}_{L^{\infty}(\mathbb{R}^{m})}\leq{\left\|u(0)\right\|}_{L^{\infty}(\mathbb{R}^{m})}\).
-
(iv)
\(\mathrm{T.V.}\left(u(t)\right)\leq\mathrm{T.V.}\left(u(0)\right)\).
Recall the following definition of moduli of continuity in time (cf. (3.54)):
The estimate (3.55) is replaced by
for a solution u of (4.23).
In several space dimensions, the Kružkov form reads
In this case, we use the test function
Here \(\omega_{\varepsilon}\) is the standard mollifier defined by
with
When v is the unique solution of the conservation law (4.25), we introduce
Kuznetsov’s lemma can be formulated as follows.
Theorem 4.5
Let u be a function in \(\mathcal{K}\), and let v be an entropy solution of (4.25). If \(0<\varepsilon_{0}<T\) and \(\varepsilon> 0\), then
where \(u_{0}=u(\,\cdot\,,0)\) and \(v_{0}=v(\,\cdot\,,0)\).
The proof of Theorem 3.14 carries over to this setting verbatim.
⋄ Example 4.6
Let us first apply this theorem to the case that u is the dimensional splitting approximation, defined with exact solution operators \(S_{\Updelta t}^{f,x}\) and \(S_{\Updelta t}^{g,y}\); cf. (4.6). We have established that \(\nu(u_{\Updelta t},\varepsilon_{0})\leq C\varepsilon_{0}\), where the constant C depends on the total variation of u 0 and the Lipschitz norm of the flux. The inequalities (4.17) and (4.18) imply
Set
In the following we always have that \(u_{\Updelta t}=u_{\Updelta t}(x,y,t)\) and \(v=v(x^{\prime},y^{\prime},s)\), although we sometimes do not indicate that, or indicate only those variables to which we would like to draw the reader’s attention. Then
where
We shall estimate \(\int I^{x}\); the estimate for I y is identical. First observe that
Therefore, if \(N{\Updelta t}=T\), then
for every function ψ. Thus
Regarding A,
Also
Hence
We have a similar estimate for the integral of I y; thus we end up with the estimate
Since we have \(v(0)=u_{\Updelta t}(0)=u_{0}\), Kuznetsov’s lemma yields
which on setting \(\varepsilon_{0}=\varepsilon=\sqrt{{\Updelta t}}\), yields
Since this estimate was obtained using the exact solution operator in each direction, there is no hope of obtaining a better estimate using numerical approximations instead of \(S_{\Updelta t}^{f,g}\). ⋄
Next, we use Kuznetsov’s lemma to estimate the rate of convergence for the front tracking approximation. This entails using a first-order (in δ) approximation to the exact solution operators, so from the previous example, the best we can hope for is that the error is bounded by \(\mathcal{O}(\delta+\sqrt{{\Updelta t}})\).
We want to estimate
where \(u=S(T)u_{0}\) and \(S_{\delta}(T)u_{0}\) denote the exact solutions of the multidimensional conservation law with flux functions f replaced by their piecewise linear and continuous approximations \(f_{\delta}\). The first term can be estimated by
while we apply Kuznetsov’s lemma, Theorem 4.5, for the second term. For the function u we choose \(u_{\eta}\), the approximate solution using front tracking along each dimension and dimensional splitting, while for v we use the exact solution with piecewise linear continuous flux functions \(f_{\delta}\) and \(g_{\delta}\), and u 0 as initial data, that is, \(v=v_{\delta}=S_{\delta}(T)u_{0}\). Thus we find, using (4.41), that
Kuznetsov’s lemma then reads
and the name of the game is to estimate \(\Lambda_{\varepsilon,\varepsilon_{0}}\).
To make the estimates more transparent, we start by rewriting \(\Lambda_{T}(u_{\eta},\phi,k)\). Since all the complications of several space dimensions are present in two dimensions, we present the argument in two dimensions only, that is, with m = 2, and denote the spatial variables by \((x,y)\). All arguments carry over to arbitrary dimensions without any change. By definition we have (in obvious notation, \(q^{f_{\delta}}(u)=\mathrm{sign}\left(u-k\right)(f_{\delta}(u)-f_{\delta}(k))\) and similarly for \(q^{g_{\delta}}\))
We now use that \(u_{\eta}\) is an exact solution in the x-direction and the y-direction on each strip \([t_{n},t_{n+1/2}]\) and \([t_{n+1/2},t_{n+1}]\), respectively. Thus we can invoke inequalities (4.17) and (4.18), and we conclude that
Observe that because we employ the projection operator π between each pair of consecutive times, we solve a conservation law in one dimension; \(u^{n+1/2}\) and u n are in general discontinuous across \(t_{n+1/2}\) and t n , respectively. The terms I 1 and I 2 are due to dimensional splitting, while I 3 and I 4 come from the projections.
Choose now for the constant k the function \(v_{\delta}(x^{\prime},y^{\prime},s^{\prime})\), and for ϕ we use Ω given by (4.46). Integrating over the new variables, we obtain
where \(I_{j}^{\varepsilon,\varepsilon_{0}}\) are given by
We will start by estimating \(I_{1}^{\varepsilon,\varepsilon_{0}}\) and \(I_{2}^{\varepsilon,\varepsilon_{0}}\).
Lemma 4.7
We have the following estimate:
Proof
We will detail the estimate for \(\left|I_{1}^{\varepsilon,\varepsilon_{0}}\right|\). Writing
we rewrite \(I_{1}^{\varepsilon,\varepsilon_{0}}\) as
with
Here we have written out all the variables explicitly; however, in the following we will display only the relevant variables. All spatial integrals are over the real line unless specified otherwise. Rewriting
we obtain
where \(\Omega^{\varepsilon}=\omega_{\varepsilon}(x-x^{\prime})\omega_{\varepsilon}(y-y^{\prime})\) denotes the spatial part of Ω.
If we rewrite \(J_{1}(t_{n+1/2},t_{n+1})\) in the same way, we obtain
and hence
Now using the Lipschitz continuity of \(q^{f_{\delta}}\), we can replace variation in \(q^{f_{\delta}}\) by variation in u, and obtain, using \(\iint\omega^{\prime}_{\varepsilon_{0}}(x-x^{\prime})\,dx\,dx^{\prime}=0\), that
using that \(\int\left|z\omega_{\varepsilon_{0}}^{\prime}(z)\right|\,dz=1\). We combine this with (4.55) to get
Inserting the estimate
we obtain
Next we consider the term J 2. We first use the Lipschitz continuity of \(q^{f_{\delta}}\), which yields
Here we integrated to unity in the variables \(s^{\prime}\) and \(y^{\prime} \), and estimated \(\int\!|\omega_{\varepsilon}^{\prime}(x\,{-}\,x^{\prime})|\kern 1.0ptdx^{\prime}\) by \(2/\varepsilon\). Finally, we used the continuity in time of the L 1-norm in the x-direction and estimated the error due to the projection. A similar bound can be obtained for \(J_{2}(t_{n+1/2},t_{n+1})\), and hence
where we used that \(\mathrm{T.V.}\left(u_{\eta}(t_{n+1/2})\right)\leq\mathrm{T.V.}\left(u_{\eta}(t_{n})\right)\). Inserting estimates (4.56) and (4.57) into (4.54) yields
where we again used that \(\mathrm{T.V.}\left(u_{\eta}\right)\) is nonincreasing. An analogous argument gives the same estimate for \(I_{2}^{\varepsilon,\varepsilon_{0}}\). Adding the two inequalities, we conclude that (4.53) holds. □
It remains to estimate \(I_{3}^{\varepsilon,\varepsilon_{0}}\) and \(I_{4}^{\varepsilon,\varepsilon_{0}}\). We aim at the following result.
Lemma 4.8
The following estimate holds:
Proof
We discuss the term \(I_{3}^{\varepsilon,\varepsilon_{0}}\) only. Recall that
The function \(u_{\eta}(x,y,t_{n}+)\) is the projection of \(u_{\eta}(x,y,t_{n}-)\), that is,
If we replace \(\iint_{\mathbb{R}^{2}}\) by \(\sum_{i,j}\iint_{I_{ij}}\) and use (4.58), we obtain
Estimating \(I_{3}^{\varepsilon,\varepsilon_{0}}(u_{\eta},v_{\delta})\) using the inverse triangle inequality, we obtain
The next step is to bound the test functions in (4.59) from above. To this end we first consider, for \(x,\bar{x}\in(i\Updelta x,(i+1)\Updelta x)\),
Integrating the time variable to unity, we easily see (really, this is easy!) that
Furthermore,
Inserting (4.60) and (4.61) into (4.59) yields
where in the final step we used that \(\mathrm{T.V.}\left(u_{\eta}(t_{n}-)\right)\leq\mathrm{T.V.}\left(u_{\eta}(0)\right)\).
The same analysis provides the following estimate for \(I_{4}^{\varepsilon,\varepsilon_{0}}(v_{\delta},u_{\eta})\):
Adding (4.62) and (4.63) proves the lemma. □
We now return to the proof of the estimate of \(\Lambda_{\varepsilon,\varepsilon_{0}}(u_{\eta},v_{\delta})\). Combining Lemma 4.7 and Lemma 4.8, we obtain
Returning to (4.49), we combine (4.50), (4.51), as well as (4.64), to obtain
Next we take the minimum over \(\varepsilon\) and \(\varepsilon_{0}\) on the right-hand side of (4.65). This has the form
The minimum is obtained for \(\varepsilon=\sqrt{b/a}\) and \(\varepsilon_{0}=\sqrt{d/c}\). We obtain
We may choose the approximation of the initial data such that \({\left\|u_{0}-u^{0}\right\|}_{L^{1}(\mathbb{R}^{2})}\!=\mathcal{O}\left(\Updelta x+\Updelta y\right)\mathrm{T.V.}\left(u_{0}\right)\). Furthermore, if the flux functions f and g are piecewise C 2 and Lipschitz continuous, then
We state the final result in the general case.
Theorem 4.9
Let u 0 be a function in \(L^{1}(\mathbb{R}^{m})\cap L^{\infty}(\mathbb{R}^{m})\) with bounded total variation, and let f j for \(j=1,\dots,m\) be piecewise C 2 functions that in addition are Lipschitz continuous. Then
as \(\eta\to 0\) when
for constants K 1 and K 2.
It is worthwhile to analyze the error terms in the estimate. We are clearly making four approximations with the front-tracking method combined with dimensional splitting. First of all, we are approximating the initial data by step functions. That gives an error of order \(\Updelta x\). Secondly, we are approximating the flux functions by piecewise linear and continuous functions; in this case the error is of order δ. A third source is the intrinsic error in the dimensional splitting, which is of order \(({\Updelta t})^{1/2}\), and finally, the projection onto the grid gives an error of order \((\Updelta x)^{1/2}\).
The advantage of this method over difference methods is the fact that the time step \({\Updelta t}\) is not bounded by a CFL condition expressed in terms of \(\Updelta x\) and \(\Updelta y\). The only relation that must be satisfied is (4.27), which allows for taking large time steps. In practice it is observed that one can choose CFL numbersFootnote 3 as high as 10–15 without loss in accuracy. This makes it a very fast method.
4.4 Operator Splitting: Diffusion
The answer, my friend, is blowin’ in the wind, the answer is blowin’ in the wind. — Bob Dylan, Blowin’ in the Wind (1968)
We show how to use the concept of operator splitting to derive a (weak) solution of the parabolic problemFootnote 4 on \(\mathbb{R}^{m}\times[0,T]\),
by solving
and
where we employ the notation \(\Updelta u=\sum_{j}u_{x_{j}x_{j}}\). We augment the equation with initial data \(u|_{t=0}=u_{0}\). Let \(S_{j}(t)u_{0}\) and \(H(t)u_{0}\) denote the solutions of (4.68) and (4.69 ), respectively, with initial data u 0. Introducing the heat kernel, we may write
Let \({\Updelta t}\) be positive and \(t_{n}=n{\Updelta t}\). Define
with the idea that u n approximates \(u(x,t_{n})\). We will show that u n converges to the solution of (4.67) as \({\Updelta t}\to 0\).
Lemma 4.10
The following estimates hold:
Proof
Equation (4.71) is obvious, since both the heat equation and the conservation law obey the maximum principle.
We know that the solution of the conservation law has the TVD property (4.72); see (4.24). Thus it remains to show that this property is shared by the solution of the heat equation. To this end, we have
which implies that
Dividing by \(\left|h\right|\) and letting \(h\to 0\), we conclude that
which proves (4.72).
Finally, we consider (4.73). We will first show that the approximate solution obtained by splitting is weakly Lipschitz continuous in time. More precisely, for each ball \(\mathcal{B}_{r}=\{x\mid\left|x\right|\leq r\}\), we will show that
for smooth test functions \(\phi=\phi(x)\), where C r is a constant depending on r. It is enough to study the case \(n_{2}=n_{1}+1\), and we set \(n_{1}=n\). Furthermore, we can write
where \(\tilde{u}^{n}=\left(S_{m}({\Updelta t})\cdots S_{1}({\Updelta t})\right)u^{n}\). This shows that it suffices to prove this property for the solutions of the conservation law and the heat equation separately. From Theorem 4.3 we know that the solution of the one-dimensional conservation law satisfies the stronger estimate
This implies that (for simplicity with m = 2)
and hence we infer that the last term of (4.75) is of order \({\Updelta t}\), that is,
The first term can be estimated as follows (for simplicity of notation we assume m = 1). Consider
Thus we conclude that (4.74) holds.
From the TVD property (4.72), we have that
Using Kružkov’s interpolation lemma (stated and proved right after this proof) we can infer, using (4.74) and (4.77), that
for all \(\varepsilon\leq\rho\). Choosing \(\varepsilon=\sqrt{\left|n_{1}-n_{2}\right|\Updelta t}\) proves the result. □
We next state and prove Kružkov’s interpolation lemma. It will be convenient to use the multi-index notation. A vector of the form \(\alpha=\left(\alpha_{1},\dots,\alpha_{m}\right)\), where each component is a nonnegative integer, is called a multi-index of order \(\left|\alpha\right|=\alpha_{1}+\cdots+\alpha_{m}\). Given a multi-index α, we define
Lemma 4.11 (Kružkov interpolation lemma)
Let \(u(x,t)\) be a bounded measurable function defined in the cylinder \(\mathcal{B}_{r+\hat{r}}\times[0,T]\), \(\hat{r}\geq 0\). For \(t\in[0,T]\) and \(\left|\rho\right|\leq\hat{r}\), assume that u possesses a spatial modulus of continuity
where \(\nu_{r,T,\hat{r}}\) does not depend on t. Suppose that for every \(\phi\in C^{\infty}_{0}(\mathcal{B}_{r})\) and \(t_{1},t_{2}\in[0,T]\),
where α denotes a multi-index.
Then for t and \(t+\tau\in[0,T]\) and for all \(\varepsilon\in(0,\hat{r}]\),
Proof
Let \(\delta\in C^{\infty}_{0}\) be a function such that
and define
Furthermore, write \(f(x)=u(x,t+\tau)-u(x,t)\) (suppressing the time dependence in the notation for f),
and
By construction, \(\sigma_{\varepsilon}\in C_{0}^{\infty}(\mathbb{R}^{m})\) and \(\mathop{\mathrm{supp}}\sigma_{\varepsilon}\subseteq\mathcal{B}_{r}\). Furthermore, \(\left|\sigma_{\varepsilon}\right|\leq 1\) and
This easily generalizes to
Next we have the elementary but important inequality
We estimate I 1 and I 2 separately. Starting with I 1, we obtain
The integrand is integrated over the domain
We further divide this set into two parts: (i) \(\left|y\right|\geq r-\varepsilon\), and (ii) \(\left|y\right|\leq r-\varepsilon\); see Fig. 4.2. In case (i) we have
since \(\sigma(y)=0\) whenever
\(\left|y\right|\geq r-\varepsilon\). In case (ii) we have
using the elementary inequality
Thus
Furthermore,
The second term I 2 is estimated by the assumptions of the lemma, namely,
Combining the two estimates, we conclude that
□
The integration domain
Next we need to extend the function u n to all times. First, define
Now let
The estimates in Lemma 4.10 carry over to the function \(u_{{\Updelta t}}\). Fix T > 0. Applying Theorem A.11, we conclude that there exists a sequence of \({\Updelta t}\to 0\) such that for each \(t\in[0,T]\), the function \(u_{{\Updelta t}}(t)\) converges to a function \(u(t)\), and the convergence is in \(C([0,T];L^{1}_{\mathrm{loc}}(\mathbb{R}^{m}))\). It remains to show that u is a weak solution of (4.67), or
for all smooth and compactly supported test functions ϕ. We have
for \(j=1,\dots,m\), where we have used that \(u^{n+(j-1)/(m+1)}\) is a solution of the conservation law on the strip \(t\in[t_{n+(j-1)/(m+1)},t_{n+j/(m+1)})\). Similarly, we find for the solution of the heat equation that
Summing (4.83) for \(j=1,\dots,m\), and adding the result to (4.84), we obtain
where
and
As \({\Updelta t}\to 0\), we have \(\chi_{j}\overset{*}{\rightharpoonup}1/(m+1)\), which proves (4.82). We summarize the result as follows.
Theorem 4.12
Let u 0 be a function in \(L^{\infty}(\mathbb{R}^{m})\cap L^{1}(\mathbb{R}^{m})\cap{BV}(\mathbb{R}^{m})\), and assume that f j are piecewise twice continuously differentiable functions for \(j=1,\dots,m\). Define the family of functions \(\{u_{{\Updelta t}}\}\) by (4.70) and (4.81). Fix T > 0. Then there exists a sequence of \({\Updelta t}\to 0\) such that \(\{u_{{\Updelta t}}(t)\}\) converges to a weak solution u of (4.67). The convergence is in \(C([0,T];L^{1}_{\mathrm{loc}}(\mathbb{R}^{m}))\).
One can prove that a weak solution of (4.67) is indeed a classical solution; see 147; . Hence, by uniqueness of classical solutions, the sequence \(\left\{u_{{\Updelta t}}\right\}\) converges for every sequence \(\left\{{\Updelta t}\right\}\) tending to zero.
4.5 Operator Splitting: Source
Experience must be our only guide; Reason may mislead us. — J. Dickinson, the Constitutional Convention (1787)
We will use operator splitting to study the inhomogeneous conservation law
where the source term g is assumed to be continuous in \((x,t)\) and Lipschitz continuous in u. In this case the Kružkov entropy condition reads as follows. The bounded function u is a weak entropy solution on \([0,T]\) if it satisfies
for all constants \(k\in\mathbb{R}\) and all nonnegative test functions \(\varphi\in C_{0}^{\infty}(\mathbb{R}^{m}\times[0,T])\).
To simplify the presentation we consider only the case with m = 1, and where \(g=g(u)\). Thus
The case in which g also depends on \((x,t)\) is treated in Exercise 4.7. Let \(S(t)u_{0}\) and \(R(t)u_{0}\) denote the solutions of
and
respectively. Define the sequence \(\{u^{n}\}\) by (we still use \(t_{n}=n{\Updelta t}\))
for some positive \({\Updelta t}\). Furthermore, we need the extension to all times, defined byFootnote 5
with
For this procedure to be welldefined, we must be sure that the ordinary differential equation (4.90) is welldefined. This is the case if g is uniformly Lipschitz continuous in u, i.e.,
For convenience, we set \(\gamma=\|g\|_{\mathrm{Lip}}\). This assumption also implies that the solution of (4.90) does not ‘‘blow up’’ in finite time, since
for some constant C g . Under this assumption on g we have the following lemma.
Lemma 4.13
Assume that u 0 is a function in \(L^{1}_{\mathrm{loc}}(\mathbb{R})\), and that u 0 is of bounded variation. Then for \(n{\Updelta t}\leq T\), the following estimates hold:
-
(i)
There is a constant M 1 independent of n and \({\Updelta t}\) such that
$$\displaystyle{\left\|u^{n}\right\|}_{L^{\infty}(\mathbb{R})}\leq M_{1}.$$(4.94) -
(ii)
There is a constant M 2 independent of n and \({\Updelta t}\) such that
$$\displaystyle\mathrm{T.V.}\left(u^{n}\right)\leq M_{2}.$$(4.95) -
(iii)
There is a constant M 3 independent of n and \({\Updelta t}\) such that for t 1 and t 2, with \(0\leq t_{1}\leq t_{2}\leq T\), and for each bounded interval \(B\subset\mathbb{R}\),
$$\displaystyle\int_{B}\left|u_{{\Updelta t}}(x,t_{1})-u_{{\Updelta t}}(x,t_{2})\right|\,dx\leq M_{3}\left|t_{1}-t_{2}\right|.$$(4.96)
Proof
We start by proving (i). The solution operator S t obeys a maximum principle, so that \({\left\|u^{n+1/2}\right\|}_{\infty}\leq{\left\|u^{n}\right\|}_{\infty}\). Multiplying (4.90) by \(\mathrm{sign}\left(u\right)\), we find that
where we have used (4.93 ). By Gronwall’s inequality (see Exercise 1.10), for a solution of (4.90), we have that
This means that
which by induction implies
Setting
proves (i).
Next, we prove (ii). The proof is similar to that of the last case, since S t is TVD, \(\mathrm{T.V.}\left(u^{n+1/2}\right)\leq\mathrm{T.V.}\left(u^{n}\right)\). As before, let u be a solution of (4.90) and let v be another solution with initial data v 0. Then we have \((u-v)_{t}=g(u)-g(v)\). Setting \(w=u-v\), and multiplying by \(\mathrm{sign}\left(w\right)\), we find that
Then by Gronwall’s inequality,
Hence,
This implies that
Inductively, we then have that
and setting \(M_{2}=e^{\gamma T}\) concludes the proof of (ii).
Regarding (iii), we know that
We also have that
where \(\left|B\right|\) denotes the length of B. Setting \(M_{3}=C+\left|B\right|C_{g}(1+M_{1})\) shows that
which implies (iii). □
Fix T > 0. Theorem A.11 implies the existence of a sequence \({\Updelta t}\to 0\) such that for each \(t\in[0,T]\), the function \(u_{{\Updelta t}}(t)\) converges in \(L^{1}_{\mathrm{loc}}(\mathbb{R})\) to a bounded function of bounded variation \(u(t)\). The convergence is in \(C([0,T];L^{1}_{\mathrm{loc}}(\mathbb{R}^{m}))\). It remains to show that u solves (4.88) in the sense of (4.87).
Using that \(u_{{\Updelta t}}\) is an entropy solution of the conservation law without source term (4.89) in the interval \([t_{n},t_{n+1/2}]\), we obtainFootnote 6
Regarding solutions of (4.90), since \(k_{t}=0\) for every constant k, we find that
Multiplying this by a test function \(\phi(t)\) and integrating over \(s\in[0,t]\), we find after a partial integration that
Since \(u_{{\Updelta t}}\) is a solution of the ordinary differential equation (4.90) on the interval \([t_{n+1/2},t_{n+1}]\) (with time running ‘‘twice as fast’’; see (4.91)), we find that
Adding this and (4.97), and summing over n, we obtain
where \(\chi_{{\Updelta t}}\) and \(\tilde{\chi}_{{\Updelta t}}\) denote the characteristic functions of the sets \(\cup_{n}[t_{n},t_{n+1/2})\) and \(\cup_{n}[t_{n+1/2},t_{n+1})\), respectively. We have that \(\chi_{{\Updelta t}}\overset{*}{\rightharpoonup}\frac{1}{2}\) and \(\tilde{\chi}_{{\Updelta t}}\overset{*}{\rightharpoonup}\frac{1}{2}\), and hence we conclude that (4.87) holds in the limit as \({\Updelta t}\to 0\).
Theorem 4.14
Let \(f(u)\) be piecewise twice continuously differentiable, and assume that \(g=g(u)\) satisfies the bound (4.92). Let u 0 be a bounded function of bounded variation. Then the initial value problem
has a weak entropy solution, which can be constructed as the limit of the sequence \(\left\{u_{{\Updelta t}}\right\}\) defined by (4.91).
4.6 Notes
Dimensional splitting for hyperbolic equations was first introduced by Bagrinovskiı̆ and Godunov 7; in 1957. Crandall and Majda made a comprehensive and systematic study of dimensional splitting (or the fractional steps method) in 52; . In 53; they used dimensional splitting to prove convergence of monotone schemes as well as the Lax–Wendroff scheme and the Glimm scheme, i.e., the random choice method. A more general introduction to operator splitting can be found in 91; .
There are also methods for multidimensional conservation laws that are intrinsically multidimensional. However, we have here decided to use dimensional splitting as our technique because it is conceptually simple and allows us to take advantage of the one-dimensional analysis.
Another natural approach to the study of multidimensional equations based on the front-tracking concept is first to make the standard front-tracking approximation: Replace the initial data by a piecewise constant function, and replace flux functions by piecewise linear and continuous functions. That gives rise to truly two-dimensional Riemann problems at each grid point \((i\Updelta x,j\Updelta y)\). However, that approach has turned out to be rather cumbersome even for a single Riemann problem and piecewise linear and continuous flux functions f and g. See Risebro 159; .
The one-dimensional front-tracking approach combined with dimensional splitting was first introduced in Holden and Risebro 93; . The theorem on the convergence rate of dimensional splitting was proved independently by Teng 178; and Karlsen 105; ; 106; . Our presentation here follows Haugse, Lie, and Karlsen 133; . Sect. 4.4 , using operator splitting to solve the parabolic regularization, is taken from Karlsen and Risebro 108; . The Kružkov interpolation lemma, Lemma 4.11, is taken from 117; ; see also 108; .
The presentation in Sect. 4.5 can be found in Holden and Risebro 95; , where also the case with a stochastic source is treated. The convergence rate in the case of operator splitting applied to a conservation law with a source term is discussed in Langseth, Tveito, and Winther 123; .
4.7 Exercises
-
4.1
Consider the initial value problem
$$\displaystyle u_{t}+f(u)_{x}+g(u)_{y}=0,\quad u|_{t=0}=u_{0},$$where f, g are piecewise twice continuously differentiable functions, and u 0 is a bounded integrable function with finite total variation.
-
(a)
Show that the solution u is Lipschitz continuous in time; that is,
$$\displaystyle{\left\|u(t)-u(s)\right\|}_{L^{1}(\mathbb{R}^{2})}\leq\left|t-s\right|\big(\|f\|_{\mathrm{Lip}}\vee\|g\|_{\mathrm{Lip}}\big)\mathrm{T.V.}\left(u_{0}\right).$$ -
(b)
Let v 0 be another function with the same properties as u 0. Show that if \(u_{0}\leq v_{0}\), then also \(u\leq v\) almost everywhere, where v is the solution with initial data v 0.
-
(a)
-
4.2
Consider the initial value problem
$$\displaystyle u_{t}+f(u)_{x}=0,\quad u|_{t=0}=u_{0},$$(4.99)where f is a piecewise twice continuously differentiable function and u 0 is a bounded, integrable function with finite total variation. Write
$$\displaystyle f=f_{1}+f_{2}$$and let \(S_{j}(t)u_{0}\) denote the solution of
$$\displaystyle u_{t}+f_{j}(u)_{x}=0,\quad u|_{t=0}=u_{0}.$$Prove that operator splitting converges to the solution of (4.99). Determine the convergence rate.
-
4.3
Prove (4.38), that is, that
$$\displaystyle\iint\left|\pi\psi-\psi\right|\,dx\,dy\leq\left(\Updelta x+\Updelta y\right)\mathrm{T.V.}\left(\psi\right),$$for all functions ψ of bounded variation.
-
4.4
Consider the heat equation in \(\mathbb{R}^{m}\),
$$\displaystyle u_{t}=\sum_{i=1}^{m}\frac{\partial^{2}u}{\partial x_{i}^{2}},\qquad u(x,0)=u_{0}(x).$$(4.100)Let \(H^{i}_{t}\) denote the solution operator for the heat equation in the ith direction, i.e., we write the solution of
$$\displaystyle u_{t}=\frac{\partial^{2}u}{\partial x_{i}^{2}},\quad u(x,0)=u_{0}(x),$$as \(H^{i}_{t}u_{0}\). Define
$$\displaystyle\begin{aligned}\displaystyle u^{n}(x)&\displaystyle=\left[H^{m}_{{\Updelta t}}\circ\cdots\circ H^{1}_{{\Updelta t}}\right]^{n}u_{0}(x),\\ \displaystyle u^{n+j/m}(x)&\displaystyle=H^{j}_{{\Updelta t}}\circ H^{j-1}_{{\Updelta t}}\circ\cdots\circ H^{1}_{{\Updelta t}}u^{n}(x),\end{aligned}$$for \(j=1,\dots,m\), and \(n\geq 0\).
For t in the interval \(\left[t_{n}+((j-1)/m){\Updelta t},t_{n}+(j/m){\Updelta t}\right]\) define
$$\displaystyle u_{{\Updelta t}}(x,t)=H^{j}_{m(t-t_{n+(j-1)/m})}u^{n+(j-1)/m}(x).$$If the initial function \(u_{0}(x)\) is bounded and of bounded variation, show that \(\left\{u_{{\Updelta t}}\right\}\) converges in \(C([0,T];L^{1}_{\mathrm{loc}}(\mathbb{R}^{m}))\) to a weak solution of (4.100).
-
4.5
We consider the viscous conservation law in one space dimension,
$$\displaystyle u_{t}+f(u)_{x}=u_{xx},\quad u(x,0)=u_{0}(x),$$(4.101)where f satisfies the ‘‘usual’’ assumptions and u 0 is in \(L^{1}\cap{BV}\). Consider the following scheme based on operator splitting:
$$\displaystyle\begin{aligned}\displaystyle U^{n+1/2}_{j}&\displaystyle=\frac{1}{2}\left(U^{n}_{j+1}+U^{n}_{j-1}\right)-\lambda\left(f\left(U^{n}_{j+1}\right)-f\left(U^{n}_{j-1}\right)\right),\\ \displaystyle U^{n+1}_{j}&\displaystyle=U^{n+1/2}_{j}+\mu\left(U^{n+1/2}_{j+1}-2U^{n+1/2}_{j}+U^{n+1/2}_{j-1}\right),\end{aligned}$$for \(n\geq 0\), where \(\lambda={\Updelta t}/{\Updelta x}\) and \(\mu={\Updelta t}/{\Updelta x}^{2}\). Set
$$\displaystyle U^{0}_{j}=\frac{1}{{\Updelta x}}\int_{(j-1/2){\Updelta x}}^{(j+1/2){\Updelta x}}u_{0}(x)\,dx.$$We see that we use the Lax–Friedrichs scheme for the conservation law and an explicit difference scheme for the heat equation. Let
$$\displaystyle u_{{\Updelta t}}(x,t)=U^{n}_{j}$$for \(\big(j-\frac{1}{2}\big){\Updelta x}\leq x<\big(j+\frac{1}{2}\big){\Updelta x}\) and \(n{\Updelta t}<t\leq(n+1){\Updelta t}\).
-
(a)
Show that this gives a monotone and consistent scheme, provided that a CFL condition holds.
-
(b)
Show that there is a sequence of \({\Updelta t}\)’s such that \(u_{{\Updelta t}}\) converges to a weak solution of (4.101) as \({\Updelta t}\to 0\).
-
(a)
Assume that u, f, and g are in \(L^{1}([0,T])\), and that g is nonnegative, while f is strictly positive and nondecreasing. Assume that
$$\displaystyle u(t)\leq f(t)+\int_{0}^{t}g(s)u(s)\,ds,\quad t\in[0,T].$$Show that
$$\displaystyle u(t)\leq f(t)\exp\Big(\int_{0}^{t}g(s)\,ds\Big),\quad t\in[0,T].$$
-
(a)
-
4.6
Assume that u and v are entropy solutions of
$$\displaystyle\begin{aligned}\displaystyle u_{t}+f(u)_{x}=g(u),&\displaystyle\quad u(x,0)=u_{0}(x),\\ \displaystyle v_{t}+f(v)_{x}=g(v),&\displaystyle\quad v(x,0)=v_{0}(x),\end{aligned}$$where u 0 and v 0 are in \(L^{1}(\mathbb{R})\cap{BV}(\mathbb{R})\), and f and g satisfy the assumptions of Theorem 4.14.
-
(a)
Use the entropy formulation (4.87) and mimic the arguments used to prove (2.60) to show that for every nonnegative test function ψ,
$$\begin{aligned}\displaystyle\iint&\displaystyle\big(\left|u(x,t)-v(x,t)\right|\psi_{t}+q(u,v)\psi_{x}\big)\,dt\,dx\\ \displaystyle&\displaystyle-\int\left|u(x,T)-v(x,T)\right|\psi(x,T)\,dx\\ \displaystyle&\displaystyle+\int\left|u_{0}(x)-v_{0}(x)\right|\psi(x,0)\,dx\\ \displaystyle\geq&\displaystyle\iint\mathrm{sign}\left(u-v\right)(g(u)-g(v))\psi\,dt\,dx.\end{aligned}$$ -
(b)
Define \(\psi(x,t)\) by (2.61), and set
$$\displaystyle h(t)=\int\left|u(x,t)-v(x,t)\right|\psi(x,t)\,dx.$$Show that
$$\displaystyle h(T)\leq h(0)+\gamma\int_{0}^{T}h(t)\,dt,$$where γ denotes the Lipschitz constant of g. Use the previous exercise to conclude that
$$\displaystyle h(T)\leq h(0)\left(1+\gamma Te^{\gamma T}\right).$$ -
(c)
Show that
$$\displaystyle{\left\|u(\,\cdot\,,t)-v(\,\cdot\,,t)\right\|}_{L^{1}(\mathbb{R})}\leq{\left\|u_{0}-v_{0}\right\|}_{L^{1}(\mathbb{R})}\left(1+\gamma te^{\gamma t}\right),$$and hence that entropy solutions of (4.98) are unique. Note that this implies that \(\left\{u_{{\Updelta t}}\right\}\) defined by (4.91) converges to the entropy solution for every sequence \(\left\{{\Updelta t}\right\}\) such that \({\Updelta t}\to 0\).
-
(a)
-
4.7
We consider the case that the source depends on \((x,t)\). For \(u_{0}\in L^{1}_{\mathrm{loc}}\cap{BV}\), let u be an entropy solution of
$$\displaystyle u_{t}+f(u)_{x}=g(x,t,u),\quad u(x,0)=u_{0}(x),$$(4.102)where g is bounded for each fixed u and continuous in t, and satisfies
$$\begin{aligned}\displaystyle\left|g(x,t,u)-g(x,t,v)\right|&\displaystyle\leq\gamma\left|u-v\right|,\\ \displaystyle\mathrm{T.V.}\left(g(\,\cdot\,,t,u)\right)&\displaystyle\leq b(t),\end{aligned}$$where the constant γ is independent of x and t, for all u and v and for a bounded function \(b(t)\) in \(L^{1}([0,T])\). We let S t be as before, and let \(R(x,t,s)u_{0}\) denote the solution of
$$\displaystyle u^{\prime}(t)=g(x,t,u),\quad u(s)=u_{0},$$for \(t> s\).
-
(a)
Define an operator splitting approximation \(u_{{\Updelta t}}\) using S t and \(R(x,t,s)\).
-
(b)
Show that there is a sequence of \({\Updelta t}\)’s such that \(u_{{\Updelta t}}\) converges in \(C([0,T];L^{1}_{\mathrm{loc}}(\mathbb{R}))\) to a function of bounded variation u.
-
(c)
Show that u is an entropy solution of (4.102).
-
(a)
-
4.8
Show that if the initial data u 0 of the heat equation \(u_{t}=\Updelta u\) is smooth, that is, \(u_{0}\in C^{\infty}_{0}\), then
$$\displaystyle{\left\|u(t)-u_{0}\right\|}_{L^{1}}\leq C\,t.$$Compare this result with (4.76).
Notes
- 1.
If we want a solution for all time, we disregard the last term in (4.4) and integrate t over \([0,\infty)\).
- 2.
We will keep the ratio \(\lambda={\Updelta t}/\Updelta x\) fixed, and thus we index only with \({\Updelta t}\).
- 3.
In several dimensions the CFL number is defined as \(\max_{i}(\left|f_{i}^{\prime}\right|{\Updelta t}/{\Updelta x}_{i})\).
- 4.
Although we have used the parabolic regularization to motivate the appropriate entropy condition, we have constructed the solution of the multidimensional conservation law independtly, and hence it is logically consistent to use the solution of the conservation law in combination with operator splitting to derive the solution of the parabolic problem. A different approach, where we start with a solution of the parabolic equation and subsequently show that in the limit of vanishing viscosity the solution converges to the solution of the conservation law, is discussed in Appendix B.
- 5.
Essentially replacing the operator H used in operator splitting with respect to diffusion by R in the case of a source.
- 6.
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Holden, H., Risebro, N.H. (2015). Multidimensional Scalar Conservation Laws. In: Front Tracking for Hyperbolic Conservation Laws. Applied Mathematical Sciences, vol 152. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47507-2_4
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