Applcaton of Resdual Dstrbuton (RD) schemes to the geographcal part of the Wave Acton Equaton Aron Roland Insttute for Hydraulc and Water Resources Engneerng, Technsche Unverstät Darmstadt, Germany Abstract The framework of Resdual Dstrbuton schemes (RD) was appled to the geographcal advecton part of the Wave Acton Equaton (WAE). The schemes proved robustness, accuracy and effcency n many dfferent applcatons, rangng from Open Ocean, Coastal Seas and Laboratory Experments. The schemes are of st up to 2 nd order n space and tme. Here the theoretcal framework for these schemes s gven and numercal experments have been carred out for certan cases where an analytcal soluton can be easly constructed.. Introducton Unstructured mesh methods have ganed n the spectral wave modellng communty strong nfluence snce the early works of Benot et al. (996), Ardhun (200) and Lau (200). Followng ths a varety of unstructured mesh methods have been ntroduced, mostly for the Euleran form of the WAE. The method s used are mostly from the class of FE (Fnte Elements) or FV (Fnte Volume). As alternatve to actually avalable numercal schemes for the soluton of the geographcal part n the WAE on unstructured meshes, Resdual Dstrbuton Schemes (RD; also known as Fluctuaton Splttng Schemes ) have been consdered here. The RD-schemes are a qute new and lovely famly of numercal schemes whch borrow deas from the FE and FV framework. As a result, compact schemes and accurate solutons can be acheved on the framework of the phlosophy of fluctuaton splttng. Abgrall (2006) gves a recent revew on the hstory and future trends of fluctuaton splttng schemes. The Resdual Dstrbuton technque was frst ntroduced by Roe (982) and further mproved by many other scentsts (e.g. Abgrall, Deconnck, Roe, Hubbard and others). 2. Resdual Dstrbuton (RD) Schemes As a startng pont, the RD-phlosophy wll be brefly ntroduced startng from the lnear advecton equaton. The lnear advecton equatons for the WAE or for any scalar quantty n a dvergence free flow reads: N t + c N = 0 () X X ECMWF Workshop on Ocean Waves, 25-27 June 202 7
ROLAND, A. APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES Ths form of the advecton equaton s vald for deep water waves that are not nteractng wth the bottom or the ambent currents. The dervaton of the RD-schemes for the flux form of the WAE (eq. 2) s shown herenafter. N t ( N ) 0 + c = (2) X X Eq. can be ntegrated by solatng the tme dervatve and ntegratng over the whole doman Ω. N da = c NdA, g t () Ω Ω where the doman Ω s dvded nto a trangular mesh wth a conformal trangulaton (see Fgure and 2). A cell s defned wth ts vertces that are numbered counter clock wse and have unque coordnates (x,y). The dscrete solutons are stored at the vertces of the trangulaton. Eq. can be rewrtten as the sum of the ntegrals over each trangular cell N. Ω N da = t N = T N da t (4) The ntegral on the rght hand sde of eq. 4 can be reformulated wth the ad of eq. as T N da = c NdA =ΦT (5) t X T Here the ntegral of the tme dervatve over the cell s equal the fluctuaton of that cell (cf. Tomach, 995). The total fluctuaton of a cell or the resdual of a cell s named ΦT. For evaluaton of these ntegrals over a trangular cell a dstrbuton functon wthn a cell for the unknown quantty must be assumed. In the fluctuaton-splttng framework, usually a lnear varaton of the dependent varable wthn the trangular cell s assumed resultng n a lnear bass functon n every trangular element. The approxmate soluton of the unknowns can be expressed as N ( ) = w,, (, ) N( ), (6) txy xy t = 8 ECMWF Workshop on Ocean Waves, 25-27 June 202
ROLAND, A.: APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES Fgure : Example of a typcal trangulated doman Ω (here the Medterranean Sea). Fgure 2: Typcal element patch (thnk black lne) and correspondng medan dual cell (red thck lne). ECMWF Workshop on Ocean Waves, 25-27 June 202 9
ROLAND, A. APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES where w are the lnear bass functons (see Fgure ) defned at each vertex. The RD-schemes borrows at ths stage the dea from the Fnte Element Method, see e.g. Donea (985), eq. 6 s smlar to hs eq. 7. The total cell fluctuaton (eq.5) can now be wrtten n a dscrete form ntroducng the spatal dervatve of wave acton, whch s gven for a lnear bass functon, w wthn the element as: N = Nn (7) 2 S T = Substtutng eq.7 n the rght hand sde of eq.5 one can rewrte the total fluctuaton as follows: Φ = T T c NdA X = λ N n = N λ n = kn = 2 = 2 = (8) ST s the area of the trangle and n the edge normal vector defned accordng to Fgure. The vectors n are scaled wth the edge length to opposte to ts vertces. λ s the lnearzed advecton vector of the cell. In ths specal case t was assumed that t s constant n spatal space (no currents, deep water) so t follows that: λ = c = c for c c (9) X, X, X, X, X = ( ) Fgure Defnton of the edge normal vectors (adapted from Tomach, 995). 20 ECMWF Workshop on Ocean Waves, 25-27 June 202
ROLAND, A.: APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES Introducng the scalars k k = λn (0) 2 one can see that the sum of all vectors n s zero so that followng dentty s vald: = n = 0. () From eq. mmedately follows that k = 0 (2) = These relatonshps are mportant for the further dervaton of the RD-schemes. The total fluctuaton of a trangular cell for the lnear conservaton law can now be rewrtten usng the above-defned relatonshps as follows:, () Φ = kn = α Φ = Φ T T T, = = = α are the redstrbuton coeffcents, whch must be n ther sum unty n order to guarantee conservaton, α = (4) = S s the medan dual cell area (see Fgure 2), whch can be evaluated as the sum of one thrd of the cell areas A connected to the certan vertex. S N con. = A = (5) The numercal approxmaton of eq. s then obtaned through the followng explct fnte volume type tme ntegraton procedure. N t = N + Φ (6) n+ n T, S T, D In order to update the soluton, the contrbuton of all fluctuatons Φ,T, that are members of the set of trangles D (element patch) connected to node I, are cumulated. The cumulated cell fluctuatons are weghted by the area of the medan dual cell, whch results n an updatng scheme, whch s very smlar to cell-vertex FV-schemes (e.g. Qan et al. 2007). However, one advantage of the above descrbed updatng procedure s that for FV-schemes the nodal update s calculated as the sum of the edge fluxes defnng the medan dual cell, whch are twce more than cells needed for the update n the FS scheme. The sum of all nodal fluctuatons, contrbutng to the nodal update, vanshes when eq.6 reaches ts steady state soluton: ECMWF Workshop on Ocean Waves, 25-27 June 202 2
ROLAND, A. APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES Φ T, = 0 (7) T, T After the total fluctuaton of a certan cell has been calculated, t must be dstrbuted over the vertces of the cell. The man problem n desgnng the schemes les n the defnton of the redstrbuton coeffcents α, whch characterzes the fnal advecton scheme. Certan desgn prncples are defned n order to develop proper redstrbuton schemes. These crtera are Conservaton or property (C) requres that the soluton at the new tme level n+ conserves the depended varable. Ths s guaranteed as long as the dstrbuton coeffcents α n eq.4 are n ther sum unty. Postvty of the scheme, or the so called property (P). For a lnear scheme, the soluton at the new tme level can be wrtten as sum of the product between the coeffcents c, resultng from the k dscretzaton of a certan scheme. The values of the updated soluton are postve f the coeffcents c k are postve N n+ n = k k k = u c u (8) For explct schemes ths condton s normally strctly connected to a CFL stablty crteron, whch must be mantaned n order to get stable and monotone solutons. For the explct schemes, n the RD-framework ths crteron s: + k t = maxt, D A (9) Lnear Preservaton (LP); Condton LP prescrbes that the numercal scheme mantans 2nd order accuracy at steady state n smooth regons of the soluton whle retanng monotoncty n the vcnty of shocks as long the CFL-lke condton s met. Based on the Godunov's order barrer theorem lnear schemes cannot be both (P) and (LP) at the same tme accordng to, whch states that lnear monotone schemes cannot be of second order wthout producng oscllatve solutons. It s possble to construct second order lnear preservng schemes, but ths makes a nonlnear redstrbuton of the fluctuatons necessary. In the framework of explct tme ntegraton schemes, ths can be done wth the applcaton of so called Flux lmters or wth the ad of blendng of lnear monotone and nonlnear non-monotone schemes (e.g. Roe, Leonard and many others). In the framework of Fnte Volume schemes, ths s called Total Varaton Dmnshng crteron. The second order fluxes of the non-monotone schemes are lmted near strong gradents, where these schemes would otherwse produce non-postve values, and monotone frst order schemes are used. The art of desgnng such lmters s to apply them only n stuatons where the second order schemes lead to non-monotone results mantanng second order accuracy as often as possble. In the RD-framework, smlar technque can be appled as done n e.g. by Hubbard & Roe (2002). One advantage of explct ntegraton schemes s that they are faster than mplct schemes because no lnear equaton system has to be solved. Another advantage s that nonlnear schemes can be desgned Lnear schemes, n the sense, that the soluton at the new tme level s a lnear combnaton of the old values of the soluton. 22 ECMWF Workshop on Ocean Waves, 25-27 June 202
ROLAND, A.: APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES n order to fulfl all the above-mentoned desgn crtera. The results are non-oscllatory monotone hgher order space/tme schemes wth a hgh accuracy. However, n certan applcatons, where a hgh spatal resoluton s needed, the lmtng factor for the explct schemes becomes the stablty crteron of eq. 9. The numercal scheme can become unfeasble wth respect to the needed computatonal tme. Implct schemes have wth respect to ths a clear advantage, they can be desgned n such a way that ther stablty and monotoncty does not depend on the CFL crteron. The prce to pay s the soluton of a lnear equaton system that evolved n the soluton procedure. The computatonal tme for the mplct scheme s roughly two tmes greater than for the mplct schemes for the case of the explct and mplct N-scheme (see Roland, 2009). In order to obtan an uncondtonally monotone mplct scheme that s also LP must lead, because of the Godunov theorem, to a nonlnear scheme (see e.g. Abgrall & Mezne, 2004 and Rchuotto et al., 2005). The resultng equaton system becomes n ths case naturally also nonlnear and teratve methods have to be used for the soluton. For the WAE ths s a crux snce the advecton part must be solved for each spectral component that average around 800-200 quanttes wth dfferent advecton veloctes makes an applcaton of a nonlnear equaton solver to a computatonally very expensve task. In ths thess, the author dd not consder nonlnear mplct schemes because of the necessary teratve soluton procedure and the assocated computatonal costs. As long as the mplct schemes are lnear and monotoncty should be retaned the schemes cannot be second order n space and tme and though LP. Hgher order lnear schemes wll always be lmted by a CFL as condton for whch monotoncty s retaned. However, the author focuses n ths work on lnear and non-lnear explct schemes as ths was the demand of the centre. 2.. Explct Resdual Dstrbuton schemes 2... The CRD N-scheme The bass of all RD-schemes s the N-Scheme, whch has ts name because of ts narrow numercal stencl as t uses only the nearest neghbour nodes to compute the nodal update of the soluton. In order to descrbe the N-Scheme for the general nonlnear conservaton law, frst the scheme for a lnear conservaton law gven by eq. 20 wll be derved. The standard N-scheme can be desgned ntroducng an upwndng for the dstrbuton of the total fluctuaton over the nodes of the elements. In order to do so, t must be dstngushed between two and one-sded nflow trangles (e.g. Tomach, 995). The amount of nflow sdes and the upwnd nodes are defned through the sgns of the k- values. For the smple case of one nflow sde, the whole fluctuaton Φ T s sent to the upwnd node. For the more complcated case of two nflow sdes, the total fluctuaton s splt between the nodes. The method of ths splttng defnes the fnal character of the scheme. The total fluctuaton can be rewrtten usng eq.2, for the case of two nflow edges as shown n Fgure 4, as ( ) ( ) Φ T = k2 N2 N + k N N. (20) ECMWF Workshop on Ocean Waves, 25-27 June 202 2
ROLAND, A. APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES Fgure 4: Two and one sde nflow wthn a trangle; taken from Tomach (995) From eq. 20 the splttng s obvous and the redstrbuton of the fluctuatons for the N-scheme reads Φ = 0, T ( ) ( ) Φ = k N N 2, T 2 2 Φ = k N N, T (2) Followng e.g. Abgrall (2006) the N-scheme can be wrtten n a more compact form ntroducng postve and the negatve k s k k + = max( k,0) = mn( k,0) (22) so that k = k + k (2) + and the upwnd resdual becomes N = n k N (24) = wth = k = n (25) The resultng splttng for the total fluctuaton may now be wrtten as + Φ T, = k ( N N ) (26) 24 ECMWF Workshop on Ocean Waves, 25-27 June 202
ROLAND, A.: APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES It can be easly seen that, when cumulatng all nodal fluctuatons accordng to eq. 27, the total fluctuaton of the cell s obtaned whch renders the scheme a conservatve one. Φ = Φ T T, (27) = The above set of equatons defnes the standard N-scheme for lnear conservatons laws accordng to eq.. The resultng scheme s vald only for the WAE n the case of the deep-water waves wthout ambent currents when the group veloctes are dvergence free n geographc space. The flux form, whch s vald n the general case, s defned as N t ( N ) 0 + c = (28) X X and can be rewrtten usng the product rule gvng N t N + X ( cxn) = + cx XN + N Xc X (29) t Snce the last term of the rght hand sde of eq. 29 do not vansh 2 n the general case, there s no obvous lnearzaton for eq. 28 and therefore the conservatve form must be solved gven through eq. 28. In order to construct a conservatve scheme n terms of the RD-framework, Csk et al (2002) ntroduced the CRD-scheme (Conservatve contour Integral based Resdual Dstrbuton scheme) where the total cell fluctuaton s evaluated over the cell contour ntegratng an arbtrary flux functon F wth a hgher order ntegraton method for the evolvng Gauss ntegral. Φ T = nds (0) T For the WAE, the flux functon s defned as = c N () X For the case of the conservatve form of the WAE, whch s vald for general nonlnear conservaton laws, the nodal resdual must be calculated accordng to Csk et al. (2002). The authors suggested replacng eq. (24) wth the followng formula n order to conserve an arbtrary flux functon F. N = n k + N ΦT (2) = Eq. (2) can easly be derved when rewrtng eq.8 for the case of the two nflow edges and calculatng the contrbuton of the upwnd node, but wth the dfference that the total fluctuaton Φ T s an unknown quantty. Ths leads drectly to eq. whch s equvalent to eq. 2. 2 The soluton of the lnearzed equatons n shallow water would result e.g. n an absence of the wave shoalng. ECMWF Workshop on Ocean Waves, 25-27 June 202 25
ROLAND, A. APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES N = k N + k N Φ () ( ) 2 2 k T If the upwnd contrbuton s calculated ths way t can be seen, when cumulatng all contrbutons from the nodes of each element accordng to eq.0, that the total fluctuaton s conserved when usng eq.2. Snce the upwnd contrbuton s now evaluated of terms of the total fluctuaton Φ T, ntegratng eq.0 numercally, the conservaton of arbtrary flux functons F s enforced. The lnearzed state, defned through the average velocty n the element, s used only for the dentfcaton of the upwnd drecton. Ths procedure can be seen as a correcton of the lnear advecton scheme for the presence of an addtonal nonlnear flux. The total cell fluctuaton can be calculated wth ad of the Smpson ntegraton. The Gauss ntegral over the trangle edges (eq.0) must be evaluated wth a hgher order ntegraton scheme (e.g. Smpson Rule) because f frst order schemes, such as the Trapezodal Rule, are used, the resultng scheme wll not accurately conserve the fluctuatons of the cell and may generate greater negatve values n the vcnty of large gradents n the soluton. The flux at the mddle of each edge can be defned as the product of the average edge normal veloctes and wave acton denstes at the nodes of each edge. T nds njds (4) j T j= lj Φ = = The Smpson rule for one edge reads: l j a+ b nds = F ( a) + 4F + F ( b) 6 2 (5) l F are the edge normal fluxes at the begnnng and the endpont of the edge and are defned as: ( ) F a = c N ( ) a, j F b = c N b, j a b (6) a and b are representng the begnnng and the endpont of the edge and the ndex j runs over the edges of the trangle. N a and N b are the wave actons and c and c are the advecton veloctes a, j b, j normal to the edge j at pont a and b respectvely. The average wave-acton-flux at the mddle of the edge j s defned as wth the average values of the wave acton and the normal advecton velocty at each node of edge j, ths s the only way to allow for a weakly non-lnear varaton along the edge of each trangle. c c a+ b + F N + N = 2 2 2 a, j b, j a b (7) The author s thankful for the dscusson wth Herman Deconnck wth respect to ths ssue. 26 ECMWF Workshop on Ocean Waves, 25-27 June 202
ROLAND, A.: APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES Ths can be rewrtten as ( ) ( ),, b,, a+ b 4F = Na c + c + N c + c 2 a j b j a j b j (8) Usng eq. 6 and eq. 7 one can rewrte eq. 5 as: l j ( ( 2 ) ( 2 a ),, b,, ) l nds = N c + c + N c + c (9) j 6 a j b j a j b j For all edges of a certan element ths can be wrtten n a dscrete form ntroducng node numbers, + and +2 for a, b and c as l ( 2 ) ( 2 ), +, + 2,, l nds = N c + c + c + c + 6 6 T l ( 2 ) ( 2 ) +,2 + 2,2, +, l + + + + 6 6 l l ( 2 + ) + ( 2 + ) + 2,, +,2 + 2,2 6 6 2 N+ c c c c 2 N+ 2 c c c c (40) Wth some algebra eq.40 can be expressed as Φ = nds = N δ (4) T T = wth δ δ δ l ( 2c c ) ( 2c c ), +, + 2,, l 6 6 l 6 6 l 6 6 = + + + l ( 2c c ) ( 2c c ) +,2 + 2,2, +, 2 2 = + + + l ( 2c c ) ( 2c c ) + 2,, +,2 + 2,2 2 = + + + (42) The coeffcent δ depends on the veloctes at the edges and the geometry of the trangle. Usng eq. 4 and eq. 4, eq. 2 can be rewrtten as N = n k N Φ = + T = n N k = + ( δ ) + = n k N δn = = (4) ECMWF Workshop on Ocean Waves, 25-27 June 202 27
ROLAND, A. APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES In ths way, the upwnd fluctuaton can be expressed only wth the nodal values that are functons of the geometry and the wave knematcs. Ths s mportant for the dervaton of the mplct FS schemes. The CRD-N scheme has very smlar characterstcs as the standard N-Scheme. It s as explct as the frst order space/tme scheme that s monotone under the CFL condton gven n eq. 9 and conservatve. However, the presented varant of the N-scheme s quas-postve, snce for the CRD-approaches postvty cannot be proven due to the numercal ntegraton of the flux functon along the edges (Csk et al. 2002). However, the resultng scheme s monotone and the negatve values are neglectable and do not alter n a practcal sense the conservaton of the scheme when set to zero. The above set of dscrete equatons descrbes the CRD-N scheme as t s mplemented n the WWM II, WW-III and WAM. 2..2. The CRD-LDA (Low Dffuson Approxmaton) scheme The LDA (Low Dffuson Approxmaton) scheme s of frst order n tme, second order n cross flow drecton and frst order n longtudal flow drecton. The scheme s LP, but snce t s lnear, t s not postve and therefore non-monotone due to the Godunov theorem. However, the LDA-scheme s used n combnaton wth lower order schemes (e.g. N-scheme) to desgn nonlnear schemes whch fulfl all the above-mentoned desgn crtera such as the PSI (Postve Streamlne Invarant) scheme or the FCT scheme usng resdual dstrbuton. The nodal fluctuaton of the standard LDA scheme reads for a certan element (e.g. Abgrall, 2006): Φ = nk + Φ (44) T, T It s easy to see that the above gven scheme s conservatve snce the sum of the redstrbuton coeffcents s unty when cumulated over all nodes of the element: + nk = = = α = (45) In order to formulate a contour ntegraton based LDA scheme, the total fluctuaton n eq. 44 must be replaced by eq. 4 n order to solve the conservatve form of the WAE. The LDA scheme s a hgher order lnear scheme and therefore not postve and non-monotone. The resultng scheme acheves frst order accuracy n tme and second order n cross flow drecton and frst order n longtudal flow drecton. 2... The CRD-PSI-scheme As gven n the ntroducton of ths chapter, a scheme, whch s conservatve, postve, and lnear preservng, must be, due to Godunov s Theorem, a nonlnear scheme. The PSI-scheme fulfls these demands and t was constructed wth the ad of a blendng parameter, whch reduces the contrbuton of the hgher order scheme when non-monotone solutons are present (e.g. Abgrall, 200). The nodal fluctuaton of the PSI scheme can be defned wth the blendng parameter accordng to Abgrall (2002) as: ( l) Φ = l Φ + Φ (46) T,, PSI T,, N T,, LDA 28 ECMWF Workshop on Ocean Waves, 25-27 June 202
ROLAND, A.: APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES wth ( φ( ) φ( 2) φ( ) ) l = max r, r, r (47) and r ϕ Φ = Φ ( x) T,, N, T, LDA x f x <, = x 0 else, (48) The resultng scheme s nonlnear and t was shown by several authors that the scheme satsfes the above defned desgn crtera. Usng the CRD-N scheme and the CRD-LDA scheme, the resultng CRD-PSI holds for any conservaton law as the one gven through eq. 2. The scheme s frst order n tme and t s at ts best second order n cross flow drecton and frst order n longtudal flow drecton. The scheme s postve for a lnear conservaton law. The PSI scheme was used herenafter to construct a truly second order space-tme scheme, whch retans the above-defned desgn crtera on the foundaton of the CRD approach of Csk et al., 2002. 2..4. The CRD-FCT-LW-PSI scheme Hubbard & Roe (2000) combned two fluctuaton splttng schemes, namely the PSI scheme presented above and the non-monotone hgher order Lax-Wendroff RD-scheme n order to desgn a monotone and postve scheme. In ths thess, the author used the concept of Csk et al. (2002) to formulate a contour ntegraton based verson of the Resdual Dstrbuton (CRD) Flux Corrected Transport scheme (CRD-FCT-LWPSI). The Lax-Wendroff scheme n the RD context reads: t Φ = + k Φ T, T 2A (49) Ths was acheved wth a generalzed FCT approach n the context of the RD-framework. The scheme s wrtten n a form, whch solates the cell contrbuton and lmts the contrbuton of the nonmonotone scheme n an optmal way near dscontnutes. Ths s done n an optmzed way n order to retan the hgher order soluton as often as possble. The FCT approach n the FS context can be descrbed n four basc steps. Frst, the hgher (HEC) and lower order contrbutons (LEC) from the Lax-Wendroff and PSI scheme are computed and the dfference of the node wse contrbuton of each scheme s estmated and defned as the so called AEC (Ant dffusve Element Contrbuton): LAX PSI AEC =Φ Φ (50) T, T, In the 2nd step the low order soluton s calculated usng eq.6 whch gves then N t PSI, n+ n PSI = N + Φ (5) T, S T, T ECMWF Workshop on Ocean Waves, 25-27 June 202 29
ROLAND, A. APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES In the thrd, most mportant step, the AEC must be corrected n such a way that the soluton at the new tme level s monotone. Ths s acheved wth correcton factors β leadng to: AEC = β AEC (52), corr The fnal update s obtaned by advancng n tme and addng the corrected AEC to the soluton of the lower order scheme accordng to: t N = N + AEC (5) n+ PSI, n+ corr, S T, T The man problem n such an approach s to formulate the correcton factors β whch must also guarantee conservaton n the case of the RD-approach. The procedure for the calculaton of the correcton factors s descrbed n detal n Hubbard & Roe (2000) and wll not be repeated for the sake of brevty. The author has also consdered n ths thess another non-monotone scheme, whch s the 2nd order upwnd control volume (UCV) scheme of Pallere (995) that reads: k Φ T, = + Φ k + = T (54) The scheme was also ncorporated n the FCT approach, but the results have been not as good as for the scheme suggested by Hubbard & Roe. As the FCT scheme n ths thess was constructed on the foundaton of the above descrbed CRD schemes, the total fluctuaton n was calculated numercally ntegratng the flux functon over the element edges. The resultng scheme should be conservatve for arbtrary flux functons. In fact the author dd not fnd any stuaton where the opposte occurs and the schemes are used as well n the TIMOR (Tdal Morphodynamcs, Zanke, 2002) for the soluton of the tracer transport and the Exner equaton. 2.2. Verfcaton of the advecton schemes 2.2.. The rotatng cylnder In order to verfy the numercal schemes the rotatng cylnder test case was nvestgated, whch was also used by Hubbard & Roe (2002) and used n many studes to verfy the mplementaton and the dffuson characterstcs of the scalar advecton schemes. In a quadratc doman Ω = [-,-] [, ] the advecton velocty vector s gven as: y λ = x (55) Ths results n a crcular current feld n whch the ntal state s equvalent to the analytcal soluton after one rotaton. The ntal dstrbuton of the unknown s dscontnuous at the edges of the cylnder (see Fg.5) and defned as follows: 0 ECMWF Workshop on Ocean Waves, 25-27 June 202
ROLAND, A.: APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES, f r 0.25 N = 0, else (56) wth ( 0.5) 2 2 2 r = x+ + y (57) The doman s resolved usng a conformal trangulaton wth 465 and 900 elements. For the explct schemes the ntegraton tme step was set to 0.0 s resultng n a CFL number wth a maxmum value of 0.997 at the boundary of the doman. The maxmal and mnmal values of the soluton for the whole smulaton and for the results after one rotaton are shown n Table. It can be seen that the hgher order non-monotone schemes (CRD-LDA. CRD-LAX and CRD-UCV) exhbt negatve values durng one rotaton. Fgure 5: Left: Intal values, analytcal soluton after one revoluton and computatonal mesh. Rght: Velocty dstrbuton accordng to eq.55. The CRD-LAX scheme result has pronounced negatve values and unwanted maxma after one evoluton. The CRD-LDA and the CRD-UCV scheme lead to moderate negatve values but are much more dsspatve snce the peak value after one revoluton s reduced to 0.6 and 0.55 respectvely. The monotone lnear N-scheme s the most dsspatve one snce half of the peak values s lost after on revoluton. The monotone nonlnear CRD-PSI scheme s nearly as dsspatve as the CRD-N scheme wth a maxmum of 0.52 after on rotaton. Clearly, the best results are obtaned wth the CRD-FCT scheme. The maxmum s preserved and the negatve values occurrng are neglectable 4. CFL =.0 Numercal Scheme CRD-N CRD-LDA CRD-LAX CRD-UCV CRD-PSI CRD-FCT Mn.value 0.00-0.4-4.00-0.2 0.00 0.00 Max. value.00.2.65..00.00 Mn. after one rotaton 0.00-0.06-0.6-0.0 0.00 0.00 Max. after one rotaton 0.49 0.6.44 0.55 0.52.00 Table : Mnmum and maxmum values of the soluton durng one rotaton and after one rotaton for the explct RD-schemes. 4 The smallest negatve value for the CRD-FCT scheme for ths test case was -8.67E-9 ECMWF Workshop on Ocean Waves, 25-27 June 202
ROLAND, A. APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES The results of the numercal smulaton for the cylnder after one revoluton usng the explct RDschemes are plotted n Fg.8. The analytcal soluton equals n that case the ntal one. The results show that the CRD-N-Scheme s the most dffusve one. The numercal dffuson n cross and longtudal drecton s consderable. The CRD-LDA scheme reduces the dffuson n cross flow drecton and mantans much better the maxmum, but the non-monotone character leads to negatve values n the soluton (e.g. negatve wave acton). The CRD-LAX scheme results n even stronger negatve values than the CRD-LDA scheme, the orgnal dstrbuton s dstorted, and the soluton s oscllatve at the wake of the cylnder. Moreover, there s a knd of phase lag n the soluton. The central scheme by Pallere (995) has very smlar characterstcs as the CRD-LDA scheme. Fgure 6: Comparson of the explct RD-schemes for a rotatng cylnder. At the top left: CRD-NScheme, top rght: CRD-LDA scheme, mddle left: CRD-Lax-Wendroff-scheme, mddle rght: CRD-UVC-scheme, bottom left: CRD-PSI-scheme and at the bottom rght the CRD-FCT-FS scheme. Note: Values greater or smaller then the colour scale are not plotted and leaved blank. 2 ECMWF Workshop on Ocean Waves, 25-27 June 202
ROLAND, A.: APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES The CRD-PSI scheme shows the combnaton of the results of the CRD-N and CRD-LDA scheme. No negatve values but a hgher cross dffuson than wth the CRD-LDA scheme though smaller than wth the CRD-N-scheme. The best results have been obtaned wth CRD-FCT scheme, whch mantans the maxmum values also after one revoluton. The phase lag of the CRD-LAX scheme s reduced, but the ntal dstrbuton undergoes a deformaton durng one revoluton. In the wake of the cylnder the CRD-LAX soluton can stll be dentfed, but the successful reducton of the AEC fluxes results n a postve soluton n that regon. 2.2.2. Shelterng by Islands One major beneft of usng genunely unstructured meshes s the flexble dscretzaton of the doman, whch makes t possble to dscretze even very tny slands and take n ths way n a very elegant way the shelterng nto account. Of course on global scale n Deep Ocean, where structured models are stll far more effcent and better tested, obstructon maps are the most economc soluton to take ths nto account, but once moved from sub-grd scale to a dscrete representaton t s of nterest to nvestgate the accuracy of the soluton method for the case of shelterng due to slands. The analytcal soluton s here clear and smple, behnd the sland there should be zero wave acton snce there s no dffuson part. The only cross dffuson ntroduced s the one by the numercal method. Below n Fgure 7 a smple unstructured mesh was desgned wth an sland n the mddle. The depth s constant 0km and the wave boundary condton s smply descrbed as all energy (Hs = m) was put n one bn at a carrer frequency of 0. Hz. On the rght hand sde of Fgure 7 the soluton, where the whte colour ndcated m sgnfcant wave heght and the shelterng s shadowng of the slands s then shown by greyscales where black means Hs = 0m. The results reveal that the deal soluton s far from beng met by all schemes; however, the nfluence of the numercal stencl n tme and space s clearly vsble. Fgure 7: Shelterng by Islands; left: numercal mesh; rght top: N-Scheme, rght mddle: PSI- Scheme, rght bottom: LF scheme. ECMWF Workshop on Ocean Waves, 25-27 June 202
ROLAND, A. APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES Fgure 8: Islands shelterng n the Medterranean Sea; dealzed case. From top to bottom: N- Scheme; PSI-Scheme and LW-Scheme If there would be some buoys just n the shelterng regon, where the varaton due to the numercal schemes s pretty hgh, numercs would have a major mpact on the soluton. In addton to shelterng 4 ECMWF Workshop on Ocean Waves, 25-27 June 202
ROLAND, A.: APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES also the penetraton depth of wave energy between slands can be a major source of error. Above (Fgure 8) a smlar case as the dealzed one s shown by just puttng same wave boundary condtons as used above n the mddle of the Medterranean Sea headng eastward towards Crete (mesh from Fgure ). The shelterng of Crete s clearly vsble as well as the nfluence of the numercal schemes. Another one qute nterestng feature n ths soluton s the penetraton depth of wave energy between the Islands of Rhodes and Karpathos. It can be clearly seen that on one hand numercal dffuson results n too lttle shelterng or wave energy but on the other hand wave energy may not even reach certan places due to dffuson where hgher order schemes stll do show penetraton of wave energy. 2.2.. Shoalng Valdaton of the flux term (2nd term at the rght hand sde of eq. 29) can be easly done by prescrbng a smple bathymetry wth lnear slope n ths case rangng from 20.m up to 0.m (the mesh ends at a depth of 0.m n the last row of elements at the shore; Fgure 9) n shallow water at the southern boundary we prescrbe smlar boundary condtons as above (Hs = m, freq. = 0.Hz) travellng perpendcular to the depth lnes. The analytcal soluton s smply gven by solvng the flux equaton n d, whch results n terms of Hs c =, (58) 2 g, Hs Hs, c g where the ndex ndcates ncdence wave condton. All three dscretzaton methods have been compared and the results are shown n terms of relatve errors n percent. Err Hs Hs, analytcal = 00 (59) H s, analytcal The results clearly show the accuracy of the method up to very shallow waters (Fgure 0) where qute hgh gradents n the soluton are present (see Fgure 0), here the results depend strongly on the dscretzaton but ths regon s n natural condtons strongly affected by wave dsspaton. It can be clearly seen that also for ths case the hgher order schemes have smaller numercal errors that reduce from 2.7% n case of the CRD-N scheme to.5% for the CRD-LW-PSI scheme. Based on these results t can be sad that the mplementaton of the schemes n valdated. In terms of computaton performance the PSI scheme s approx..5 as expensve as the N-Scheme and the LW scheme s approx. 2.8 tmes more expensve than the N-scheme. From my experence and based on the present representaton of the physcs n the models, takng nto account the lack of modellng dffracton and treatng GSE (at least on unstructured meshes) the PSI schemes s the scheme to be used. The mplementaton of the PSI scheme wthn the CRD approach usng mplct tme ntegrators wll be the next step for the author. ECMWF Workshop on Ocean Waves, 25-27 June 202 5
ROLAND, A. APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES Fgure 9: Left: numercal mesh; Rght: depth n the computatonal doman..5 CRD-N-Scheme Depth 20 8 6 4 Hs [m] 2.5 2 2 0 8 Water Depth [m] 6.5 4 2 0 0 500 000 500 2000 2500 000 500 4000 Length [m] Fgure 0: Soluton of the WAE for lnear depth profle 6 ECMWF Workshop on Ocean Waves, 25-27 June 202
ROLAND, A.: APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES 20 2.5 2 CRD-N-Scheme CRD-PSI-Scheme CRD-FCT-LWPSI-Scheme Depth 8 6 4 Error [%].5 2 0 8 Water Depth [m] 6 0.5 4 2 0 0 0 500 000 500 2000 2500 000 500 4000 Length [m] 5 2.5 2 CRD-N-Scheme CRD-PSI-Scheme CRD-FCT-LWPSI-Scheme Depth 4.5 4.5 Error [%].5 2.5 2 Water Depth [m].5 0.5 0.5 0 0 000 00 200 00 400 500 600 700 800 900 4000 Length [m] Fgure : Errors over the whole doman (top) and n the vcnty of the shore (bottom). Outlook and Concluson The formalsm of RD schemes has been ntroduced to the WAE. There s no reason why t should be only appled n geographcal space, spectral space could be treated as well but snce the two operators n spectral space are not commutng, smlar to x- and y-space advecton, the use of a splttng as done n WW s qute nce for spectral space. The problem of splttng the equaton and the ongong errors n splttng remans a problem of most of the mplementaton (as dscussed n Roland, 2009). Another possblty besdes the splttng would be, snce the RD-schemes are nherently multdmensonal, a ECMWF Workshop on Ocean Waves, 25-27 June 202 7
ROLAND, A. APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES non-splt soluton of the advecton part of the WAE on unstructured meshes, whch would actually resemble the numercs of the SWAN model but on unstructured meshes. If so, the queston of the lmter must be thoroughly nvestgated snce t s not acceptable, that for non-statonary smulatons the whole left hand sde of the equaton becomes lmted only because we cannot lnearze all source terms properly (remember: Patankar s Laws, that say that proper lnearzed st order schemes are uncondtonally stable). Moreover, the stff contrbutons of the dfferent terms may result n large Egenvalues n the matrx resultng n ll-condtoned systems. Therefore A-stable schemes need to be used and a proper numercal framework has to be mplemented. However, here are not only numercal ssues to be taken nto account but of course also physcal as e.g. wave dffracton, whch wll n a certan way result n cross dffuson of wave energy n the shadowng regon of slands. From my pont of vew t s not a proper approach to approxmate physcs by numercal errors but ths s and wll stay realty to certan extend also n the next decade. Ths statement s of course hghly dealstc but progress and physcs must go hand n hand wth mproved numercal math otherwse the remanng parameters of the partly emprcal physcal processes depend to a certan extend on the numercal schemes. References Abgrall, R. and Roe, P.L., 200. Hgh order fluctuaton schemes on trangular meshes. J. Sc. Comput., 9(-), 6. Abgrall, R., 200. Toward the ultmate conservatve scheme: followng the quest. J. Comput. Phys., 67(2), 277 5. Abgrall, R., Mezne, M., 200. Constructon of second order accurate monotone and stable resdual dstrbuton schemes for unsteady flow problems, J. Comput. Phys. 88, 6 55. Abgrall, R. Resdual dstrbuton schemes: current status and future trends. Computer and Fluds, 5(7):64--669, 2006. nvted paper Ardhun, F., 200: Swell across the contnental shelf. Ph.D. thess, Naval Postgraduate School, pp. Benot, M., F. Marcos and F. Becq. 996. Development of a thrd generaton shallow-water wave model wth unstructured spatal meshng, Proceedngs of 25th Internatonal Conference on Coastal Engneerng, ASCE, 465 478. Donea, J., 984. A Taylor-Galerkn method for convectve transport problems. Internat. J. Numer. Methods Eng., 20, 0-20. Csík, Á., Rcchuto, M. and Deconnck, H., 2002. A conservatve formulaton of the multdmensonal upwnd resdual dstrbuton schemes for general non lnear conservaton laws. Journal of Computatonal Physcs, 79(), pp 286- Hubbard, M. E., Roe, P. L., 2000. Compact hgh-resoluton algorthms for tme-dependent advecton on unstructured grds. Internat. J. Numer. Methods Fluds, 7-76. Lau, J-M. 200, A Study of Wnd Waves Hndcastng on the Coastal Waters, PhD Thess, Natonal Cheng Kung Unversty, Tanan, Tawan Qan, Q., Stefan, H., G. and Voller, V. R., 2007, A physcally based flux lmter for QUICK calculatons of advectve scalar transport Internatonal Journal for Numercal Methods n Fluds Volume 55, Issue 9, Pages: 899-95 8 ECMWF Workshop on Ocean Waves, 25-27 June 202
ROLAND, A.: APPLICATION OF RESIDUAL DISTRIBUTION (RD) SCHEMES Pallere, H., 995. Multdmensonal upwnd resdual dstrbuton schemes for the Euler and Naver- Stokes equatons on unstructured grds. PhD Thess, Unverste Lbre de Bruxelles, Belgum. Rcchuto, M., Csík, Á., Deconnck, H., 2005. Resdual dstrbuton for general tmedependent conservaton laws. J. Comput. Phys. 209 (), 249 289. Roe PL. Characterstc-based schemes for the Euler equatons. Ann. Rev. Flud Mech 986;8:7 65. Roland, A., (2009), Development of WWM II: Spectral wave modelng on unstructured meshes. Ph.D. thess, Technsche Unverst at Darmstadt, Insttute of Hraulc and Water Resources Engneerng. Tomach, G.T., 995. A genunely mult-dmensonal upwndng algorthm for the Naver-Stokes equatons on unstructured grds usng a compact, hghly-parallelzable spatal dscretzaton. PhD thess, Unversty of Mchgan, USA. Zanke, U.C.E., 2002. Hydromechank der Gernne und Küstengewässer (Parey Verlag) Acknowledgements The author s ndebted to Peter Janssen for gvng the author the opportunty to ntroduce the RD- Framework to the WAM code. The author s thankful to Jean Bdlot for sharng nsghts of the code and beng helpful n a varety of questons. The author also wsh to thank the whole centre for provdng a professonal and warm atmosphere durng my stay, especally, I lke to thank Domnque Lucas who has helped n many questons of source code development and HPCF usage. ECMWF Workshop on Ocean Waves, 25-27 June 202 9
40 ECMWF Workshop on Ocean Waves, 25-27 June 202